PRINCIPLES 


OF 


REINFORCED  CONCRETE 
CONSTRUCTION 


BY 


F.    E.    XURNEAURE 

Dean  of  the  College  of  Engineering:  University  of  Wisconsin 


AND 


E.    R.   MAURER 

Professor  of  Mechanics,   University  of  Wisconsin 


FIRS  T    EDITION 
FIRST  THOUSAND 


NEW    YORK 

JOHN  WILEY  &  SONS 

London:    CHAPMAN   &   HALL,    Limited 
1907 


K 


Copyright,  1907 

BY 
F.  E.  TURNEAURE  AND  E.  R.  MAURER 


Knbrrt  Dr«mmonl>  anb  Qlompang 


PREFACE. 


IN  the  present  volume  the  authors  have  endeavored  to 
cover,  in  a  systematic  manner,  those  principles  of  mechanics 
underlying  the  design  of  reinforced  concrete,  to  present  the 
results  of  all  available  tests  that  may  aid  in  establishing  coeffi- 
cients and  working  stresses,  and  to  give  such  illustrative 
material  from  actual  designs  as  may  be  needed  to  make  clear 
the  principles  involved. 

The  work  is  essentially  divided  into  two  parts:   Chapters 

I  to  VI  treat  of  the  theory  of  the  subject  and  the  results  of 
experiments,  while  the  remaining  chapters  treat  of  the  use  of 
reinforced  concrete  in  various  forms  of  structures.    In  Chapter 

II  the  properties  of  plain  concrete  and  of  steel  are  considered 
to  a  sufficient  extent  to  give  accurate  notions  of  their  relation 
to  the  general  subject  in  hand.     The  subjects  of  adhesion  and 
of  relative  contraction  and  expansion  are  also  discussed  in  this 
chapter.    In  Chapter  III  is  given  a  full  theoretical  treatment 
of  reinforced  concrete,  avoiding  so  far  as  possible  empirical 
rules  and  methods;  and  in  Chapter  IV  are  presented  the  most 
important  available  tests  on  beams  and  columns,  analyzed  and 
correlated,  so  far  as  may  be,  with   reference  to   theoretical 
principles.    The  subjects  of  working  stresses  and  economical 
proportions  are  considered  in  Chapter  V.    In  Chapter  VI  are 
brought   together  in  convenient   form  all  the  formulas  and 
diagrams  needed  for  practical  use.     There  are  also  included 
tables  relating  to  reinforcing  bars  and  a  comprehensive  table 

iii 


161588 


iv  PREFACE. 

of  the  strength  of  floor  slabs.  This  chapter  is,  for  most  pur- 
poses, complete  in  itself,  so  that  the  reader  need  not  refer  to 
any  other  portion  of  the  work  in  order  to  use  it  in  designing. 

Following  the  theoretical  portions  are  chapters  on  the 
application  of  reinforced  concrete  to  building  construction, 
arches,  retaining  walls,  dams,  and  miscellaneous  structures. 
In  these  chapters  the  analysis  of  various  features  is  given, 
where  the  use  of  reinforced  concrete  involves  problems  new 
and  unfamiliar.  A  complete  general  analysis  of  the  solid  arch 
rib  is  also  given,  which,  the  authors  believe,  offers  advantages 
over  the  usual  graphical  method.  It  is  primarily  an  analytical 
method,  but-  may  be  shortened  by  obvious  simple,  graphical 
aids.  Stresses  in  the  concrete  and  steel  are  readily  calculated 
by  the  use  of  diagrams  in  Chapter  VI.  In  the  chapters  on  the 
application  of  reinforced  concrete  it  has  not  been  the  aim  to 
cover  practical  construction  in  all  its  phases;  for  this  the 
reader  is  referred  to  the  more  voluminous  works  on  the  subject. 
It  is  hoped,  however,  that  as  a  treatment  of  the  principles  of 
design  the  work  may  prove  of  service  to  the  student  and  the 
engineer. 

F.   E.   TURNEAURE. 

E.  R.  MAURER. 
MADISON,  Wis.,  Sept.,  1907. 


CONTENTS. 


CHAPTER   I. 

PAGE 

INTRODUCTORY 1 

Historical  Sketch.     Use  and  Advantages  of  Reinforced  Concrete. 


CHAPTER    II. 
PROPERTIES  OF  THE  MATERIAL 8 

Concrete:  General  Requirements.  Cement.  Sand.  Broken 
Stone  or  Gravel.  Proportion  of  Ingredients.  Consistency. 
Compressive  Strength.  Tensile  Strength.  Transverse  Tests, 
Shearing  Strength.  Elastic  Properties.  Stress-strain  Curve  in 
Compression.  Modulus  of  Elasticity.  Elastic  Limit.  Stress- 
strain  Curves.  Coefficient  of  Expansion.  Contraction  and 
Expansion.  Weight.  Properties  of  Cinder  Concrete.  Rein- 
forcing Steel :  General  Requirements.  Special  Forms  of  Bars. 
Quality  of  the  Material.  Tensile  Strength.  Modulus  of  Elas- 
ticity. Elastic  Elongation.  Coefficient  of  Expansion.  Prop- 
erties of  Concrete  and  Steel  in  Combination :  Adhesion.  Me- 
chanical Bond.  Ratio  of  Moduli  of  Elasticity.  Tensile  Stress 
and  Elongation  of  Concrete  when  Reinforced.  Relative  Con- 
traction and  Expansion. 

CHAPTER   III. 
GENERAL  THEORY 43" 

Kinds  of  Members.  Relation  of  Stress  Intensities  in  Concrete 
and  Steel.  Distribution  of  Stress  in  a  Homogeneous  Beam. 

v 


vi  CONTENTS. 

PAGE 

Purpose  and  Arrangement  of  Steel  Reinforcement.  The  Com- 
mon Theory  of  Flexure  and  its  Modification  for  Concrete. 
Resisting  Moment  and  Inefficiency  of  Concrete  Beams.  Varieties 
of  Flexure  Formulas.  Flexure  Formulas  for  Working  Loads. 
Flexure  Formulas  for  Ultimate  Loads.  Flexure  Formulas  for 
any  Load  up  to  Ultimate.  Comparison  of  Flexure  Formulas. 
Flexure  Formulas  for  T-Beams.  Beams  Reinforced  for  Com- 
pression. Flexure  and  Direct  Stress.  Shearing  Stresses  in  Re- 
inforced Beams.  Bond  Stress.  Strength  of  Columns. 


CHAPTER   IV. 
TESTS  OF  BEAMS  AND  COLUMNS.        .        .        .        .        .113 

Beams :  Methods  of  Failure  of  a  Reinforced  Concrete  Beam. 
Tests  of  Beams  giving  Steel- tension  Failures.  Position  of 
Neutral  Axis  and  Value  of  n.  Observed  and  Calculated 
Stresses  in  the  Steel.  Compressive  Stresses  in  Concrete.  Con- 
clusions Regarding  Moment  Calculations.  Tests  in  which 
Failure  Occurred  by  Diagonal  Tension.  Methods  of  Web  Rein-  ^ 
forcement.  Action  of  Web  Reinforcement.  Effect  of  Stirrups. 
Results  of  Tests.  Tests  on  T-Beams.  Conclusions  as  to 
Shearing  Strength.  Beams  Reinforced  in  Compression.  Col- 
umns :  Plain  Concrete  Columns.  Tests  on  Columns  with  Lon- 
gitudinal Reinforcement.  Hooped  Concrete  Columns.  Fatigue 
Tests. 


CHAPTER   V. 
WORKING  STRESSES  AND  GENERAL  CONSTRUCTIVE  DETAILS     166 

Working  Stresses  and  Factors  of  Safety.  Relative  Effect  of 
Dead  and  Live  Loads.  Beams :  Working  Formulas.  Working 
Stresses  in  Concrete  and  Steel.  Quality  of  Steel.  Bond  Stress. 
Shearing  Stresses.  Calculation  of  Web  Reinforcement.  Spacing 
of  Bars  Economical  Proportions  and  Working  Stresses.  T- 
Beams.  Columns  :  Working  Stresses.  Economy  in  the  Use  of 
Reinforced  Columns.  Use  of  Steel  of  High  Elastic  Limit.  Use 
of  Steel  at  Ordinary  Working  Stresses.  Durability  of  Rein- 
forced Concrete  :  Protection  of  Steel  from  Corrosion.  Fireproof- 
ing  Effect  of  Concrete.  Reinforcing  Against  Shrinkage  and 
Temperature  Cracks. 


CONTENTS.  vii 

CHAPTER   VI. 

PAGE 

FORMULAS,  DIAGRAMS  AND  TABLES     .        .  .        .     197 

Rectangular  Beams.  T-Beams.  Beams  Reinforced  for  Com- 
pression. Flexure  and  Direct  Stress.  Shearing  and  Bond 
Stress.  Columns.  Shrinkage  and  Temperature  Stresses.  Co- 
efficients and  Working  Stresses.  Tables. 

CHAPTER   VII. 

BUILDING  CONSTRUCTION      . 237 

Division  of  the  Subject.  General  Arrangement  of  Concrete 
Floors.  Stresses  in  Continuous  Beams.  Effect  of  Rigid  Sup- 
ports. Slabs  Reinforced  in  Two  Directions.  Reinforcement  to 
Prevent  Cracks.  Floor  Slabs  Supported  on  Steel  Beams. 
Floor  Slabs  in  All-Concrete  Construction.  Beams  and  Girders. 
Columns.  Examples  of  Floor  and  Column  Design.  Footings. 
Walls  and  Partitions. 

CHAPTER   VIII. 

ARCHES 263 

Advantages  of  the  Reinforced  Arch.  Methods  of  Reinforce- 
ment. Analysis  of  the  Arch:  General  Method  of  Procedure. 
Thrust,  Shear,  and  Moment  at  the  Crown.  Thrust,  Shear,  and 
Moment  at  Any  Point.  Partial  Graphical  Calculation.  General 
Observations.  Division  of  Arch  Ring  to  give  Constant  ds/I. 
Temperature  Stresses.  Stresses  Due  to  Shortening  of  Arch  from 
Thrust.  Deflection  of  the  Crown.  Unsymmetrical  Arches. 
Applications.  Maximum  Stresses  in  the  Arch  Ring.  Illustra- 
tive Examples  of  Design. 

CHAPTER    IX. 

RETAINING-WALLS  AND  DAMS      ....  .     289 

Advantages  of  Reinforced  Concrete.  Retaining  Walls :  Method 
of  Determining  Stability.  Equivalent  Fluid  Pressure  for 
Ordinary  Masonry  Walls.  Stability  of  Reinforced  Concrete 
Walls.  Design  of  Wall.  Illustrative  Examples.  Rect- 
angular Walls  Supported  at  the  Top.  Dams:  Stability  and 
Examples. 


viii  CONTENTS. 

CHAPTER   X. 

PA  OK 

MISCELLANEOUS  STRUCTURES 304 

Simple  Beam  Bridges.  Concrete  Trestles.  Pipe  and  Box 
Culverts.  The  Circular  Culvert.  The  Rectangular  Culvert. 
Arrangement  of  Reinforcement.  Illustrative  Examples.  Con- 
duits and  Pipe  Lines.  Tanks,  Reservoirs,  Bins,  etc. 


REINPORCED-CONCRETE   CONSTRUCTION. 

CHAPTER  I. 
INTRODUCTORY. 

i.  Historical  Sketch. — The  invention  of  reinforced  con- 
crete is  usually  credited  to  Joseph  Monier,  but  his  first  con- 
structions are  antedated  by  those  of  Lambot,  who  in  1850 
constructed  a  small  boat  of  reinforced  concrete  and  in  1855 
exhibited  the  same  at  the  Paris  Exposition.  In  this  latter  year 
Lambot  took  out  patents  on  this  form  of  construction;  it  was 
regarded  by  him  as  especially  well  adapted  to  shipbuilding, 
reservoir  work,  etc. 

In  1861,  Monier,  who  was  a  Parisian  gardener,  constructed 
tubs  and  tanks  of  concrete  surrounding  a  framework  or  skeleton 
of  wire.  In  the  same  year  Coignet  announced  his  principles 
for  reinforcing  concrete,  and  proposed  construction  of  beams, 
arches,  pipes,  etc.  Both  he  and  Monier  executed  some  work  in 
the  new  material  at  the  Paris  Exposition  of  1867.  In  this  year 
Monier  took  out  patents  on  his  reinforcement.  It  consists  of 
two  sets  of  parallel  bars,  one  set  at  right  angles  to  and  lying 
upon  the  other,  thus  forming  a  mesh  of  bars.  This  system,  and 
slight  modifications  of  it,  are  extensively  used  at  the  present 
time,  particularly  for  slab  reinforcement.  Though  even  the 
early  Monier  patents  covered  principles  of  wide  application, 
still  the  early  work  in  reinforced  concrete  was  confined  to  a 
comparatively  narrow  field. 


2  INTRODUCTORY.  [On.  I. 

In  1884-5  the  German  and  American  rights  of  the  Monier 
patents  fell  into  the  hands  of  German  engineers.  One  of  these, 
G.  A.  Wayss,  and  J.  Bauschinger  at  once  began  an  experimental 
investigation  of  the  Monier  system,  and  in  1887  they  published 
their  findings.  The  investigation  proved  reinforced  concrete  a 
valuable  means  of  construction,  and  furnished  some  formulas 
and  methods  for  design.  From  this  time  on,  the  use  of  re- 
inforced concrete  in  Austria  spread  rapidly,  and  a  few  years 
ago  the  engineers-  of  that  country  were  credited  with  having 
done  more  to  develop  the  new  construction  than  those  of  any 
other  country.  Among  these  engineers  should  be  mentioned 
Melan,  who  in  the  early  90's  originated  a  system  in  which  I  or 
T  beams  are  the  principal  element  of  strength,  providing  com- 
pressive  as  well  as  tensile  strength.  In  Germany  government 
regulations  hindered  the  application  of  reinforced  concrete  for  a 
time,  but  now  it  is  widely  used  in  that  country.  Over  two 
hundred  systems  of  reinforcement,  it  has  been  stated,  have 
been  developed  in  Germany  alone. 

In  France  the  Monier  system  was  never  developed  as  in 
countries  already  mentioned.  Here,  as  elsewhere,  many  other 
systems  of  reinforcement  were  invented  from  time  to  time, 
among  which  should  be  mentioned  that  of  Hennebique,  who 
was  probably  the  first  to  use  stirrups  and  abent-up"  bars. 
This  system  is  in  general  use,  and  the  elements  of  Hennebique's 
system  are  probably  more  widely  used  than  those  of  any 
other. 

In  England  and  America  the  first  use  of  iron  or  steel  with 
concrete  arose  in  the  effort  to  fireproof  the  former  by  means  of 
the  latter.  Attempting  to  utilize  also  the  strength  of  concrete, 
Hyatt  built  beams  of  concrete  reinforced  with  metal  in  various 
ways,  and  with  Kirkaldy  of  London  performed  tests  on  such 
beams  and  published  the  results  of  the  investigation  in  1877. 
The  first  reinforced-concrete  work  in  the  United  States  was  done 
in  1875  by  W.  E.  Ward,  who  constructed  a  building  in  New  York 
state  in  which  walls,  floor-beams,  and  roof  wrere  made  of  con- 
crete reinforced  with  metal  to  provide  tensile  strength.  But 


§  1.]  HISTORICAL  SKETCH.  3 

the  Pacific  Coast  saw  the  actual  early  development  of  this  form 
of  construction.  H.  P.  Jackson,  G.  W.  Percy,  and  E.  L.  Ran- 
some  were  the  pioneer  workers.  Jackson  has  been  credited  with 
reinforced  constructions  dating  as  far  back  as  1877,  but  Ransome 
executed  the  most  notable  early  examples.  Among  these  are  a 
warehouse  (1884  or  '85),  a  factory  building  a  few  years  later,  the 
building  of  the  California  Academy  of  Science  (1888  or  '89), 
and  the  museum  building  of  Leland  Stanford  Junior  University 
(1892).  Percy  was  the  architect  of  the  last  two,  The  museum 
building  contains  spans  of  45  feet  and  is  reinforced  through- 
out. This  and  the  Academy  building  withstood  the  recent 
earthquake  remarkably  well — the  museum  better  than  its  two 
brick  annexes. 

Other  pioneer  constructors  in  reinforced  concrete  in  this 
country  were  F.  von  Emperger  and  Edwin  Thacher.  The 
former  introduced  the  Melan  system  (1894)  and  built  the  first 
reinforced  arch  bridges  of  considerable  span.  Thacher  also 
was — and  still  is — a  bridge-builder.  His  first  large  reinforced- 
concrete  bridge  was  built  in  1896  and  was  without  precedent 
here  or  in  Europe. 

America  is  the  home  of  the  " patent  bar".  Both  Ransome 
and  Thacher  invented  bars  known  by  their  respective  names, 
the  patented  feature  of  which  is  to  furnish  a  "grip"  between 
bar  and  concrete;  besides  these  two  there  are  several  others  on 
the  market  designed  to  give  additional  grip  or  bond.  There  are 
also  patented  bars  for  supplying  " shear  reinforcement".  Some 
of  these  forms  have  been  introduced  into  Europe. 

Reinforced-concrete  construction  has  had  a  remarkable 
development,  particularly  in  the  last  decade,  and  is  now  re- 
garded by  engineers  and  architects  generally  as  a  safe  form  of 
construction  with  a  wide  field  of  economical  application.  Com- 
mon practice  has  already  established  itself  in  some  directions, 
and  rational  principles  are  available  for  much  design  work. 
Outstanding  uncertainties  are  under  investigation  in  many 
quarters,  and  the  time  is  not  far  distant  when  "good  practice" 
in  reinforced  concrete  will  have  been  established. 


4  INTRODUCTORY.  [Cn.  I- 

2.  Use  and  Advantages  of  Reinforced  Concrete. — A  com- 
bination of  steel  and  concrete  constitutes  a  form  of  construc- 
tion possessing  to  a  large  degree  the  advantages  of  both  mate- 
rials without  their  disadvantages.  It  will  be  desirable  at  the 
outset  to  consider  briefly  these  advantages  in  order  better  to 
appreciate  the  field  in  which  this  type  of  construction  is  likely 
to  be  most  successful. 

Steel  is  a  material  especially  well  suited  to  resist  tensile 
stresses,  and  for  such  purposes  the  most  economical  form — 
the  solid  compact  bar — is  well  adapted.  To  resist  compressive 
stresses  steel  must  be  made  into  more  expensive  forms,  con- 
sisting of  relatively  thin  parts  widely  spread,  in  order  to  provide 
the  necessary  lateral  rigidity.  A  serious  disadvantage  in  the 
use  of  steel  in  many  locations  is  its  lack  of  durability;  and, 
again,  a  comparatively  low  degree  of  heat  destroys  its  strength, 
thus  rendering  it  necessary  to  add  a  protective  covering  where 
a  fire-proof  structure  is  demanded.  Steel  is  a  relatively  expen- 
sive building  material,  and  its  cost  tends  to  increase. 

Concrete  is  characterized  by  low  tensile  strength,  relatively 
high  compressive  strength,  and  great  durability.  It  is  a  good 
fire-proof  material,  and  therefore  serves  as  a  good  fire-proof 
covering  for  steel.  It  is  also  found  that  steel  well  covered  by 
concrete  is  thoroughly  protected  from  corrosion.  Concrete  is 
also  a  comparatively  cheap  material  and  is  readily  available 
in  almost  any  location. 

In  the  design  of  structural  members  these  qualities  of  steel 
and  concrete  will  lead  to  the  use  of  the  two  materials  about 
as  follows:  For  those  structural  members  carrying  purely  ten- 
sile stresses  steel  must  be  employed,  but  it  may  be  surrounded 
by  concrete  as  a  protection  against  corrosion  and  fire,  or  merely 
for  the  sake  of  appearance.  For  those  members  sustaining 
purely  compressive  stresses  concrete  is  fundamentally  the 
better  and  cheaper  material.  With  concrete  costing  30  cents 
per  cubic  foot,  for  example,  and  steel  4  cents  per  pound,  or 
about  $20.00  per  cubic  foot,  and  with  working  stresses  of 
400  and  15,000  lbs/in2,  respectively,  the  relative  cost  of  the 


§2.]  USE  AND   ADVANTAGES. 

30    .  2000 

two  materials  for  carrying  a  given  load  is  as  -TT^.  is  to 


400  15,000' 

or  as  45  is  to  80.  For  large  and  compact  compressive  members 
plain  concrete  will  therefore  naturally  be  used,  especially  where 
durability  is  a  factor.  For  more  slender  members,  however, 
such  as  long  columns,  plain  concrete  is  too  brittle  a  material, 
and  therefore  too  much  affected  by  secondary  and  unknown 
stresses  to  be  satisfactory;  and  for  such  members  steel  alone, 
or  the  two  materials  in  combination,  will  preferably  be  used. 
Steel  may  be  used  with  concrete  in  the  form  of  small  rods 
to  reinforce  the  concrete;  or  it  may  be  used  in  larger  sec- 
tions and  simply  surrounded  and  held  rigidly  in  place  by 
the  concrete,  most  of  the  load  being  carried  by  the  steel;  or, 
finally,  a  steel  column  may  be  used  and  merely  fireproofed 
by  the  concrete.  As  the  cost  of  steel  in  the  form  of  rods  is 
much  less  than  in  the  form  of  built  members,  and  as  com- 
pressive stresses  can,  in  general,  be  carried  more  cheaply  by 
concrete  than  by  steel,  economical  construction  will  lead  to 
the  use  of  the  maximum  amount  of  concrete  and  the  minimum 
amount  of  steel  consistent  with  safety,  although  this  prin- 
ciple will  be  modified  by  various  practical  considerations. 

For  those  structural  forms  in  which  both  tension  and  com- 
pression exist,  that  is  to  say,  in  all  forms  of  beams,  the  com- 
bination of  the  two  materials  is  particularly  advantageous. 
Here  the  tensile  stresses  are  carried  by  steel  rods  embedded 
in  the  concrete  near  the  tension  side  of  the  beam.  The  steel 
is  thus  used  in  its  cheapest  form,  it  is  thoroughly  protected 
by  the  concrete,  and  the  compressive  stresses  are  carried  by 
the  concrete.  Concrete  alone  cannot  be  used  to  any  appre- 
ciable extent  to  carry  bending  stresses  on  account  of  its  low 
and  uncertain  tenacity,  but  a  concrete  beam  with  steel  rods 
embedded  in  it  to  carry  the  tensile  stresses  is  a  strong,  economical, 
and  very  durable  form  of  structure. 

From  these  considerations  it  follows  that  reinforced-con- 
crete  construction  is  advantageous  to  varying  degrees  in  dif- 
ferent types  of  structures.  Some  of  the  most  important  of 


6  INTRODUCTORY.  [Cn  I. 

these  types  will  here  be  noted,  together  with  the  advan- 
tages accompanying  the  use  of  reinforced  concrete  in  their 
design. 

3.  Buildings. — This  type  of  construction  is  especially  useful 
for  floor-slabs  and  to  a  somewhat  less  degree  for  beams,  girders, 
and  columns.     It  is  also  well  adapted  for  footings  in  founda- 
tions, being  more  economical  than  I-beam  footings  embedded 
in  concrete. 

4.  Culverts  and  small  Girder  Bridges. — Very  satisfactory  on 
account  of  its  simplicity  and  economy  as  compared  to  masonry 
arches,   and  because  of    its  durability   as   compared  to  steel 
bridges. 

5.  Retaining-walls,  Dams,  and  Abutments. — Often  economical 
for  such  structures  as  compared  to  ordinary  masonry.     Plain 
masonry  structures  of  this  kind  are  designed  to  resist  lateral 
forces  by  their  weight  alone,  the  resulting  compressive  stresses, 
except  in  extremely  large  structures,   being  very  small  and 
much  below  safe  values.     By  the  use  of  reinforced  concrete 
these  structures  can  be  designed  of  a  more  economical  type 
and  so  arranged  as  to  utilize  the  concrete  in  the  form  of  beams, 
thus  developing  more  nearly  the  full  compressive  strength  of 
the  material.    The  steel  reinforcement  is  fully  protected  from 
corrosion,  a  factor  which  prevents  the  use  of  all-steel  frames 
for  structures  of  this  class. 

6.  Arch  Bridges. — In  this  form  of  structure  reinforced  con- 
crete possesses  less  advantage  over  ordinary  masonry  than  in 
those  forms  where  the  compressive  stresses  are  less  important. 
In  an  arch  the  stresses  are  principally  compressive,  and  these 
do  not  require  steel  reinforcement;    it  is  only  to  provide  for 
the  relatively  small  bending  stresses  due  to  moving  loads,  or 
as  a  precaution  against  undesirable  cracks,  that  steel  is  ser- 
viceable.    No  large  economy  can  be  obtained  through  its  use. 
By  reason  of  greater  simplicity  and  the  less  expensive  abutments 
required,  a  flat-top  culvert  or  beam  bridge,  with  abutments 
of  reinforced  concrete,  is  more  advantageous  for  short  spans 
than  the  arch. 


§  11.]  USE  AND   ADVANTAGES.  7 

7.  Reservoir  Walls,  Floors,  and  Roofs. — Very  well  adapted 
as  a  durable  material  and  lending  itself  to  lighter  design  than 
common  masonry. 

8.  Conduits  and  Pipe  Lines. — Reinforced  concrete  can  often 
be  used  to  great  advantage  in  a  water-conduit  or  large  sewer. 
It  is  also  sometimes  used  for  pipe  lines  and  tanks  under  pres- 
sure, the  steel  being  relied  upon  to  resist  the  tensile  stresses, 
while  the  concrete  serves  as  a  protection  and  as  a  water-tight 
covering.     The  amount  of  steel    may  thus  be  determined  by 
considerations  of  strength  alone,  where  otherwise  a  much  larger 
amount  of  metal  would  be  needed  and  in  a  more  expensive 
form. 

9.  Elevated  Tanks,  Bins,  etc. — Advantageous  because  of  its 
durability  and  its  adaptability  in  the  construction  of  heavy 
floors   and   walls   subjected   to   lateral   pressure.     Of   especial 
value  for  coal-bins,  either  for  flooring  and  lining  alone,  or  for 
the  entire  structure. 

10.  Chimneys  and  Towers. — Possesses  advantages  over  brick 
or  stone  masonry  in  the  fact  that  it  forms  a  structure  of  mono- 
lithic character,  resulting  in  greater   certainty  in  the  stresses 
and  economy  in  design. 

11.  Piles,  Railroad  Ties,  etc. — The  use  of  a  moderate  amount 
of  steel  with  concrete  so  as  to  give  to  this  material  a  reliable 
tensile  and  bending  resistance  has  opened  the  way  for  its  use 
in  a  great  variety  of  forms,  not  only  as  complete  structures, 
or  important  members  of  structures,  but  also  in  many  special 
individual  forms.     Concrete  piles  are  valuable  substitutes  for 
piles  of  wood  where  the  latter  would  be  subject  to  deteriora- 
tion.    Reinforced-concrete  ties  offer  some  evident  advantages 
over  ties  of  wood  or  steel.    This  material  is  also  well  adapted 
to  many  other  special  uses,  particularly  where  durability  is- 
an  important  factor. 


CHAPTER  II. 

PROPERTIES   OF   THE   MATERIALS. 

I2»  In  a  design  where  two  or  more  materials  are  combined 
in  the  same  member  the  stresses  in  the  different  materials 
depend  upon  the  elastic  properties  as  wrell  as  upon  the  super- 
imposed loads.  Therefore  in  making  such  designs  a  knowl- 
edge of  these  elastic  properties  is  quite  as  necessary  as  a  knowl- 
edge of  the  strength  of  the  materials. 

CONCRETE. 

13.  General  Requirements. — The  conditions  to  be  met  in 
reinforced-concrete  construction  require  the  use,  generally, 
of  a  concrete  of  relatively  high  grade.  In  this  type  of  con- 
struction the  strength  of  the  material  is  of  much  greater  im- 
portance than  it  is  in  many  forms  of  plain  concrete  design, 
as  the  dimensions  of  the  structures  are  more  directly  dependent 
upon  strength  and  less  upon  weight.  A  comparatively  strong 
concrete  is  therefore  found  to  be  economical. 

It  is  especially  important,  also,  that  the  concrete  be  of 
uniform  quality  and  free  from  voids,  as  the  sections  are  com- 
paratively small  and  the  stability  of  the  structure,  to  a  much 
greater  extent  than  is  the  case  with  massive  concrete,  is  de- 
pendent upon  the  integrity  of  every  part.  Thoroughly  sound 
concrete  is  also  required  in  order  to  insure  good  adhesion  to 
the  steel  reinforcement  and  adequate  protection  of  the  steel 
from  corrosion  and  from  fire.  These  requirements  call  for 
great  care  in  the  preparation  and  placing  of  the  material. 


§  16.]  PROPERTIES  OF  CONCRETE.  9 


Concrete  is  subject'  to  great  variations  in  its  properties, 
owing  t^he  great  variations  in  the  character  and  proportions 
of  its  i^Bdients  and  in  its  preparation.  It  is  therefore  diffi- 
cult to^idge  from  results  of  tests  made  under  certain  con- 
ditions as  to  what  may  fairly  be  expected  of  a  concrete  pre- 
pared under  other  conditions;  so  that  it  is  very  important 
that  regular  and  systematic  tests  of  the  material  as  actually 
used  be  made  during  the  progress  of  the  work. 

14.  Cement. — Portland   cement   only  should  be  used;    it 
should  meet  such  standard  specifications  as  those  of  the  Ameri- 
can Society  of  Civil  Engineers.     The  rapidity  of  hardening  of 
different  cements  varies   considerably  and  may  be  an  element 
requiring  special  attention  where  the  structure  is  to   receive 
its  load  very  early  or  where  such  load  is  to  be  long  deferred. 

15.  Sand. — The  sand  should  be  free  from  clay  and  pref- 
erably of  coarse  grain.      A  fine  sand    requires  more   cement 
than  a  coarse  sand  for  equal  strength,  and  more  water  for  a 
like  consistency.      In  the  case  of  a  very  fine  sand  the  differ- 
ence may  be  very  marked,  so  that  unless  care  is  taken  and 
special  tests  made,  the  resulting  concrete  is  likely  to  be  porous 
and  deficient  in  strength  and  adhesive  power.     Where  the  use 
of  fine  sand  is  contemplated,  tests  of  strength  may  show  that 
a  considerable  extra  cost  may  be  justified  in  securing  a  coarser 
material.     The  effect  of  size  of  sand  is  shown  in  Art.  19. 

1 6.  Broken  Stone  and  Gravel. — Both  materials  are  satis- 
factory, but  they  should  be  screened  to  remove  the  dust  or 
sand  and  to  remove  particles  larger  than  the  maximum  size 
desired.     Beyond  this,  the  screening  of  stone  to  size  is  unde- 
sirable unless  an  artificial  mixture  is  to  be  made,  as  it  tends  to 
increase  the  proportion  of  voids.     Gravel  may  be  sufficiently 
uniform  in  quality  so  that  the  sand  need  not  be  removed,  but 
it  will  usually  require  screening  in  order  to  insure  a  concrete 
of  definite  proportions. 

The  maximum  desirable  size  of  stone  or  gravel  depends  upon 
the  size  of  the  structural  forms  and  the  size  and  spacing  of  the 
reinforcement,  it  being  desirable  to  use  as  large  a  size  of  aggre- 


10  PROPERTIES   OF   THE  MATERIALS.  [Cn.  II. 

gate  as  will  admit  of  convenient  working.  Maximum  sizes  of 
stone  of  J  inch  to  1J  inches  are  common,  but  on  heavy  work, 
with  rods  widely  spaced,  there  is  no  objection  to  -&j$[  larger 
sizes. 

The  crushing  strength  of  a  gravel  concrete  is  usually  a 
little  less  than  one  of  broken  stone  of  the  same  proportion  of 
voids,  but  the  difference  is  unimportant.  The  difference  in 
tensile  strength  is  not  well  determined,  but  the  few  tests  avail- 
able indicate  about  the  same  relative  difference  as  in  com- 
pressive  strength. 

17.  Proportions  of  Ingredients. — The  proportions  commonly 
used  vary  from  about  1:2:4  to  1:3:6  of  cement,  sand,  and 
broken  stone  respectively;    or   the  equivalent    proportions  if 
gravel  be  used.     Richer  mixtures  than  1 :2:4  are  not  common, 
nor  poorer  mixtures  than  1:2J:5,  although  with  a  well-graded 
sand  a  very  satisfactory  concrete  can  be  made  of  1:3:6  pro- 
portions.     Occasionally  where   the  design   is   determined   by 
other  considerations  than  strength  and  cost,  a  very  rich  mix- 
ture or  a  poor  one  may  be  desirable,  but  where  these  elements 
determine  the  design,  the  most  economical  concrete  will  be 
a  rich   concrete   of   about   the   proportions   above   indicated. 
Customary  proportions,  such  as  1:2:4,  should  not  be  blindly 
adopted.     In  any  important  work  a  careful  study  of  the  mate- 
rials and  of  the  best  proportions  to  use  for  economy  and  strength 
will  be  well  repaid.     To  secure  sound  and  reliable  work,  with 
good  adhesion  and  tensile  strength,  there  must  be  no  unfilled 
voids  in  the  stone  and  little  or  none  in  the  sand.     The  former 
is  of  more  importance  than  the  latter,  and  if  cost  and  strength 
are  to  be  reduced  it  should  be  done  by  using  a  poorer  mortar 
to  fill  the  voids  in  the  stone.     For  a  more  detailed  study  of 
this    subject   the    reader    is    referred    to    special    works    on 
concrete. 

1 8.  Consistency. — The  tendency  in  all  kinds  of   concrete 
construction  is  to  use  a  wetter  mixture  than  formerly.     Rela- 
tively dry  concrete  thoroughly  tamped  will  give  slightly  greater 
strength  than  a  wet  mixture;    however,   if  not  too  wet  the 


§  19.]  PROPERTIES  OF  CONCRETE.  11 

difference  is  not  great,  and  considering  the  difficulty  and  ex- 
pense of  securing  the  necessary  amount  of  tamping  of  the 
dry  mixture,  better  results  can  usually  be  secured  by  using  a 
plastic  mixture.  This  is  especially  true  with  reference  to 
obtaining  a  dense,  homogeneous  concrete.  The  usual  prac- 
tice now  is  to  make  the  consistency  such  that  the  concrete  will 
require  only  moderate  tamping  or  puddling  to  bring  the  mass 
to  a  homogeneous  condition.  Such  concrete,  while  somewhat 
weaker  than  the  ideal  compacted  concrete,  will,  under  actual 
conditions,  be  much  more  reliable  and  will  be  free  from  voids. 
In  the  case  of  reinforced-concrete  work  reliability  is  more 
important  than  maximum  strength,  and  is  promoted  by  using 
concrete  of  such  consistency  that  it  can  readily  be  worked 
into  place  in  the  forms  and  around  the  reinforcing  steel.  In 
practice  the  consistency  varies.  Some  use  a  concrete  which 
requires  considerable  tamping  and  working,  while  others  use 
a  concrete  which  will  practically  flow  into  place.  The  dryer 
the  concrete  the  closer  the  inspection  required  when  the  mate- 
rial is  placed;  on  the  other  hand  very  wet  concrete  is  not  as 
strong  and  needs  to  be  promptly  poured  to  prevent  segrega- 
tion of  the  materials. 

19.  Compressive  Strength. — The  compressive  strength  of 
concrete  is  dependent  upon  many  factors  so  that  it  is  diffi- 
cult and  at  the  same  time  somewhat  misleading  to  present 
" average  values".  Obviously,  in  any  important  work,  the 
strength  should  be  determined  under  the  actual  conditions 
under  which  the  concrete  is  used.  Uniformity  is  quite  as 
important  as  average  strength. 

One  of  the  best  series  of  tests  is  that  made  at  the  Water- 
town  Arsenal  for  Mr.  George  A.  Kimball,  Chief  Engineer  of 
the  Boston  Elevated  Railway  Company.*  The  concrete  was 
made  of  five  brands  of  Portland  cement,  coarse,  sharp  sand, 
and  broken  stone  up  to  2J-inch  size.  The  concrete  was  well 
rammed  into  the  molds,  water  barely  flushing  to  the  surface. 

*  Tests  of  Metals,  1899,  p.  717. 


12 


PROPERTIES  OF  THE  MATERIALS. 


[Cn.  II. 


The  specimens  were  buried  in  wet  ground  after  being  taken 
from  the  molds.     The  average  results  were  as  follows: 

TABLE  No.  1. 

COMPRESSIVE    STRENGTH    OF   CONCRETE. 
WATER-TOWN  ARSENAL,  1899. 


Mixture. 

B  and  of  Cement. 

Strength,  Pounds  per  Square  Inch. 

7  Days. 

1  Month. 

3  Months. 

6  Months. 

1  :  2  :  4  j 

{ 
1:3:6  < 

Saylor  

1724 

1387 
904 
2219 
1592 

2238 
2428 
2420 
2642 
2269 

2702 
2966 
3123 
3082 
2608 

3510 
3953 
4411 
3643 
3612 

Atlas  

Alpha  

Germania 

Alsen                        

Average  

1565 

1625 
1050 
892 
1550 
1438 

.  2399 

2568 
1816 
2150 
2174 
2114 

2896 

2882 
2538 
2355 
2486 
2349 

3826 

3567 
3170 
2750 
2930 
3026 

Saylor  

Atlas 

Alpha 

Germania                

Alsen   .             

Average  

1311 

2164 

2522 

3088 

In  a  series  of  tests  made  at  the  Watertown  Arsenal  for 
Mr.  George  W.  Rafter,  the  following  average  values  were 
obtained  on  concrete  about  20  months  old.*  The  voids  in 
the  broken  stone  were  practically  filled.  The  mixture  was  of 
damp-earth  consistency : 

Strength. 

4467  lbs/in2 
3731       " 


Cement. 
1 
1 
1 


Sand. 
1 

2 


2553 


Results  as  high  as  indicated  by  the  preceding  values  can- 
not be  safely  counted  upon  in  practice.  Wet  concrete  will  show 
a  lower  strength  than  concrete  as  dry  as  that  in  the  above 
tests,  especially  for  the  earlier  periods,  but  the  difference 
becomes  less  with  lapse  of  time,  and  a  fairly  soft  plastic  con- 


*  Tests  of  Metals,  1898. 


§  19.]  PROPERTIES  OF  CONCRETE.  13 

crete  will  acquire  about  the  same  strength  as  dry  concrete 
within  three  or  four  months.  A  very  wet  concrete  will,  how- 
ever, |  continue  to  be  somewhat  weaker  than  one  containing 
less  water,  and  while  such  a  concrete  may,  on  the  whole,  be 
desirable,  its  deficiency  in  strength  as  compared  to  maximum 
values  should  not  be  overlooked.  Other  variations  in  condi- 
tions, such  as  rapid  drying  out,  or  the  use  of  very  fine  sand, 
for  example,  may  give  results  materially  below  those  here 
quoted. 

The  following  average  results  of  a  large  number  of  tests 
in  the  series  made  for  Mr.  Rafter,  already  referred  to,  show 
the  relative  strengths  of  dry,  plastic,  and  wet  concrete  at  the 
age  of  about  twenty  months.  The  dry  mixtures  were  only  a 
little  more  moist  than  damp  earth  and  required  much  ramming; 
the  plastic  mixtures  required  a  moderate  amount  of  ramming 
to  bring  water  to  the  surface;  the  wet  mixtures  quaked  like 
liver  under  moderate  ramming.  Five  brands  of  cement  were 
used: 

Consistency.  Mean  Compressive  Strength. 

Dry 2348  lbs/in2 

Plastic 2203       " 

Wet 2129       " 

In  actual  practice  results  are  very  likely  to  be  less  favorable 
to  dry  mixtures  on  account  of  the  great  difficulty  of  securing 
adequate  tamping. 

The  effect  of  size  of  sand  has  been  thoroughly  investigated 
by  Feret.  Fig.  1,  from  Johnson's  "Materials  of  Construction", 
showrs  results  obtained  by  Feret  on  1 : 3  mortar  after  hardening 
one  year  in  fresh  water.  The  sand  used  consisted  of  mixtures  of 
various  proportions  of  fine  (.0  to  .5  mm.),  medium  (.5  to  2  mm.), 
and  coarse  (2  to  5  mm.)  sand,  and  in  the  figure  the  result  from 
any  particular  mortar  is  recorded  in  the  triangle  at  such  dis- 
tances from  the  three  base-lines  as  will  represent  the  propor- 
tions of  each  size  sand  used.  Lines  of  equal  strength  were 
then  drawn  in  the  diagram.  Thus  the  strength  of  the  rnortar 


14 


PROPERTIES  OF  THE  MATERIALS. 


[Cn.  II. 


in  which  only  fine  sand  was  used  was  only  1400  lbs/in2.  The 
maximum  strength  of  3500  lbs/in2  was  obtained  from  a  mix- 
ture containing  about  85%  of  coarse  sand  and  15%  of  fine, 
with  a  very  little  sand  of  medium  size.  This  diagram  shows 
in  a  striking  manner  the  effect  of  size  of  sand. 

As  illustrating,  further,  the  variation  in  results  that  may 
be  expected,  due  to  variation  in  conditions,  60-day  tests  on 


FIG.  1.— Effect  of  Size  of  Sand. 

6-inch  cubes  mixed  wet  and  left  exposed  in  a  room  are  thus 
reported  by  Professor  Talbot :  * 

1:2:4  mixture,  average  strength  1520  lbs/in2. 
1:3:6  mixture,  average  strength  1230  lbs/in2. 

These  low  results  were  probably  due  to  too  rapid  drying. 
Thirty-day  tests  on  thirty-eight  4-inch  cubes  of  1:2:4  wet 
concrete,  made  at  the  University  of  Wisconsin  in  1906,  gave 
an  average  strength  of  1785  lbs/in2,  with  many  values  below 
1500.  The  cubes  were  stored  in  air  in  a  basement  room.  The 
average  result  of  tests  on  cylinders  6  inches  in  diameter  by 
18  inches  high  of  the  same  material  was  1780  lbs/in2.  That 
a  less  strength  was  not  shown  for  the  cylinders  than  for  the 
cubes  was  probably  due  to  the  greater  effect  of  rapid  drying 

*  University  of  Illinois,  Bulletin  No.  4,  1906. 


§20.]  PROPERTIES  OF  CONCRETE.  15 

on  the  small  cubes.  A  fine  sand  was  used.  In  another  test 
of  the  same  material,  two  4-inch  cubes  cured  in  water  averaged 
1600  lbs/in2,  while  two  similar  cubes  hardened  in  air  aver- 
aged 1370.  Of  four  beams  tested  which  failed  by  crushing, 
two  which  were  cured  in  water  failed  at  an  average  load  of 
8380  Ibs.,  and  two  cured  in  air  at  a  load  of  7400  Ibs.,  thus 
showing  a  less  relative  difference  in  strength  than  in  the  case 
of  the  cubes.  Other  compression  tests  in  which  a  very  fine 
though  commonly  used  sand  was  employed  gave  results  but 
little  more  than  1000  lbs/in2  in  30  days. 

Considering  the  various  results  noted  it  may  be  concluded 
that  under  reasonably  good  conditions  as  to  character  of  ma- 
terial and  workmanship  an  average  strength  of  2000-2200 
lbs/in2  may  be  expected  of  a  1:2:4  concrete  in  30  to  60  days, 
the  rate  of  hardening  depending  upon  the  consistency  and  the 
temperature;  and  for  a  1:3:6  concrete  a  strength  of  1600  to 
1800  lbs/in2.  It  will  be  noted  that  these  values  are  but  little 
less  than  the  minimum  averages  given  in  Table  No.  1  (page  12) 
for  30-day  tests. 

It  is  important  that  the  strength  be  determined  by  actual 
tests  of  the  material  proposed  to  be  used,  and  if  the  results  are 
too  low  the  ingredients  or  proportions  should  be  modified  until  a 
satisfactory  result  is  obtained.  Where  the  usual  proportions 
give  low  results  it  will  generally  be  advisable  to  increase  the 
richness  of  the  concrete  rather  than  to  reduce  the  working 
stresses. 

20.  Tensile  Strength. —  Comparatively  few  tests  have  been 
made  on  the  tensile  strength  of  concrete.  This  property  is, 
however,  important,  and  should  receive  more  attention,  as  it 
is  closely  involved  in  the  most  common  and  most  dangerous 
type  of  failure  of  reinforced-concrete  beams.  The  tensile 
strength  of  concrete  is  usually  stated  as  approximately  one- 
tenth  to  one-eighth  of  the  compressive  strength,  although  there 
is  no  fixed  relation  between  the  two  values.  The  character 
of  the  sand  and  the  aggregate  has  probably  a  greater  influence 
on  the  tensile  strength  than  upon  the  compressive,  and  poor 


16  PROPERTIES  OF  THE  MATERIALS.  [Cn.  II. 

workmanship  undoubtedly  has.     Tests  by  Mr.  W.  H.  Henby  * 
gave  results  as  follows: 

Mixture.  Compressive  Strength.  Tensile  Strength. 

1:2:4  3000  lbs/in2  180  lbs/in2 

1:3:6  1800       "  115       " 

Tests  by  Professor  W.  K.  Hatt  f  gave  the  following  results: 

Kind  of  Concrete.         ^    Compres^ve.  Strength,     TensH^  Strength, 

1:2:4  (broken  stone)  30  311 

1:2:5  "  90  2413  359 

1:2:5  "  28  2290  237 

1:5  (gravel)  90  2804  290 

1:5  28  2400  253 

Tests  by  Professor  Ira  H.  Woolsen  J  on  1:2:4  mixtures 
5  to  7  weeks  old  gave  an  average  tensile  strength  of  161  lbs/in2, 
compared  to  1753  lbs/in2  compressive  strength. 

Professor  Talbot  obtained  values  for  1:3:6  concrete  from 
50  to  84  days  old  of  178,  160,  and  170  Ibs/in2.§ 

21.  Tensile  Strength  as  Determined  by  Transverse  Tests.— 
The  transverse  strength  of  plain  concrete  depends  almost 
entirely  upon  its  tensile  strength,  although  the  modulus  of 
rupture  is  considerably  greater  than  the  strength  in  plain  ten- 
sion owing  to  the  curved  form  of  the  stress-strain  diagram. 
Feret  ||  found  a  very  nearly  constant  ratio  of  1.95  of  modulus 
of  rupture  to  tensile  strength.  The  value  of  this  ratio  will 
ordinarily  range  from  1.5  to  2.  Transverse  tests  of  different 
concretes  should  therefore  show  about  the  same  relative  results 
as  tensile  tests.  They  are  in  fact  quite  as  significant  in  this 
connection. 

Some  of  the  best  tests  on  transverse  strength  are  those 
made  by  William  B.  Fuller,  and  given  in  full  in  Taylor  and 

*  Jour.  Assn.  Eng.  Soc.,  Sept.  1900. 

f  Jour.  West  Soc.  Eng.,  Vol.  IX,  1904,  p.  234. 

j  Eng.  News,  Vol.  LIII,  1905,  p.  561. 

§  Bulletin  No.  1,  Univ.  of  111.,  1904. 

11  Etude  Exp4rimentale  du  Ciment  Arme".     Paris,  1906. 


§  22.]  PROPERTIES  OF  CONCRETE.  17 

Thompson's  work  on  Concrete.*  The  following  average  results 
were  obtained  for  33-35-day  tests. 

Mixture  by  Volume.  Average  Modulus 

of  Rupture. 

1:2.16:4.08  439  Ibs/in2 

1:2.16:5.1  380       ' ' 

1:3.24:5.1  285       " 

1:3.24:6.12  226       " 

1:3.24:7.14  239       " 

Here  we  find  the  strength  of  the  1:3.24:6.12  mixture  only 
about  one-half  that  of  the  1:2.16:4.08  mixture,  indicating  the 
relative  weakness  in  tension  of  the  lean  mixture. 

The  results  herein  given,  both  of  tensile  and  of  transverse 
tests,  indicate  that  the  quality  of  the  concrete  has  a  greater 
relative  effect  on  the  tensile  strength  than  on  the  compressive 
strength,  the  strength  of  a  1:3:6  mixture  being  not  mo  re  than 
two-thirds  that  of  a  1:2:4  mixture.  Reasonable  values  for 
ultimate  tensile  strength  would  appear  to  be  about  as  follows: 

1:2:4  mixture 160-200  lbs/in2 

1:3:6       "       100-125      " 

22.  Shearing  Strength. — There  is  a  lack  of  uniformity  among 
writers  as  to  just  what  is  meant  by  the  term  "  shearing  strength  ", 
resulting  in  a  wide  variation  in  the  suggested  values  for  working 
stresses.  In  this  work  the  authors  will  use  the  term  as  it  is 
commonly  thought  of  among  American  engineers,  to  denote 
the  strength  of  the  .material  against  a  sliding  failure  when  tested 
as  a  rivet  or  bolt  would  be  tested  for  shear;  that  is,  when  the 
maximum  shearing  stresses  are  confined  to  a  single  plane. 

Tests  made  under  the  direction  of  Professor  C.  M.  Spofford 
on  cylinders  5  inches  in  diameter  with  ends  securely  clamped 
in  cylindrical  bearings  gave  results  as  follows: 

Shearing  Compressive  Ratio  of 


Mixture. 

Strength, 
lbs/in2. 

Strength, 
lbs/in2. 

Shearing  to 
Comp.  Strength. 

1:2:4 

1480 

2350 

.63 

1:3:5 

1180 

1330 

.89 

1:3:6 

1150 

1110 

1.04 

*  Concrete,  Plain  and  Reinforced.     N.  Y.,  1906. 


18  PROPERTIES   OF  THE  MATERIALS.  [Cn.  II. 

Tests  made  at  the  University  of  Illinois  on  rectangular  speci- 
mens tested  in  a  similar  manner  gave  the  following  average 
results : 

Shearing  Compressive  Ratio  of 

Mixture.  Strength,  Strength,  Shearing  to 

lbs/in2.  lbs/in2.          Comp.  Strength. 

1:2:4  1418  3210  .44 

1:3:6  1250  2290  .57 

Tests  made  by  punching  through  plates  gave  shearing 
strengths  varying  from  37  to  90  per  cent  of  the  compressive, 
the  value  depending  upon  the  form  of  test-piece.* 

Tests  by  M.  Feret  on  mortar  prisms  gave  results  for  shear- 
ing strength  equal  to  about  one-half  the  crushing  strength. 

The  ordinary  crushing  failure  is  really  a  failure  by  shear- 
ing, and  under  such  conditions  the  crushing  stress  is,  theo- 
retically, twice  the  shearing  stress,  the  angle  of  shear  being 
45°.  Results  of  tests  give  a  somewhat  greater  inclination 
than  45°,  so  that  the  crushing  stress  is  somewhat  greater  than 
twice  the  actual  shearing  stress. 

We  may  then  conclude,  both  from  theory  and  from  tests, 
that  the  shearing  strength  of  concrete,  in  the  sense  here  used, 
is  nearly  one-half  the  crushing  strength.  It  is  in  fact  so  large 
that  it  will  need  to  be  considered  only  in  exceptional  cases. 

Some  writers  used  the  term  " shearing  stress"  to  mean 
quite  a  different  thing  from  that  discussed  above,  namely, 
the  complex  action  which  occurs  in  the  web  of  a  beam.  In 
this  case  there  exist  direct  tensile  and  compressive  stresses 
which  at  the  neutral  axis  are  equal  in  intensity  to  the  vertical 
and  horizontal  shearing  stresses.  The  limit  of  distortion  in 
the  concrete  will  be  reached,  and  failure  will  occur,  when  the 
.tensile  strength  of  the  material  is  exceeded.  Such  a  failure 
may  perhaps  be  called  a  shearing  failure,  but  is  more  strictly 
a  failure  in  tension  in  a  diagonal  direction,  and  is  so  considered 
in  this  work.  Treated  as  a  shearing  failure  the  strength  should 
be  very  nearly  the  same  as  the  tensile  strength  of  the  material 
determined  in  the  usual  way.  In  practice  the  diagonal  tensile 

*  Bulletin  No.  8,  Univ.  of  111.,  1906. 


§23.]  PROPERTIES  OF  CONCRETE.  19 

stresses  in  a  beam  must  often  be  considered,  but  shearing 
stresses,  as  such,  will  be  dangerous  only  in  exceptional  circum- 
stances, such  as  exist  where  a  heavy  load  is  applied  close  to 
a  support. 

-^  23.  Elastic  Properties  of  Concrete. — Stress-strain  Curve 
'  in  Compression. — In  the  design  of  combination  structures,  such 
as  those  of  steel  and  concrete,  it  is  necessary  to  know  the 
relative  stresses  under  like  distortions.  These  will  depend 
upon  the  moduli  of  elasticity  of  the  two  materials.  For  purposes 
of  safe  design  we  need  to  know  also  the  elastic-limit  strength 

Fig.  2  represents  typical  stress-strain  curves  for  concrete 
in  compression.  Curves  C,  D,  E,  and  F  were  obtained  at  the 
University  of  Wisconsin  from  tests  on  cylinders  6  inches  in 
diameter  by  18  inches  high.  The  concrete  was  1:2:4  limestone 
concrete  30  days  old.  The  ultimate  strengths  ranged  from  1500 
to  2300  lbs/in2.  Curves  A  and  B  are  typical  curves  selected 
from  the  Watertown  Arsenal  tests  already  quoted,  and  repre- 
sent 1:2:4  and  1:3:6  concrete  respectively. 

Unlike  the  elastic  line  for  steel,  the  line  for  concrete  is 
slightly  curved  almost  from  the  beginning,  the  curvature  grad- 
ually increasing  towards  the  end.  There  is,  however,  no  point  of 
sharp  curvature  as  for  ductile  materials.  A  release  of  load  at 
a  moderate  stress,  such  as  500  to  600  lbs/in2,  will  usually  show 
a  small  set  indicating  imperfect  elasticity.  A  second  applica- 
tion of  the  load  will,  however,  give  a  straighter  line  than  the 
first  and  there  will  be  much  less  permanent  set  following  the 
release  of  load.  After  a  few  repetitions  of  load  there  will  be 
no  further  set  and  the  stress-strain  line  will  become  a  straight 
line  up  to  the  load  applied.  There  is  a  limit  of  stress,  however, 
beyond  which  repeated  applications  of  load  will  continue  to 
add  to  the  permanent  deformation  and  the  specimen  will 
ultimately  fail.  The  general  behavior  under  repeated  stress 
is  indicated  in  Fig.  3,  from  tests  on  concrete  similar  to  those 
represented  in  Fig.  2.  For  a  very  exhaustive  study  of  this 
subject  the  reader  is  referred  to  the  work  of  Bach.* 
*  Zeit.  V.  dt.  Ing.,  1895,  etc. 


20 


PROPERTIES  OF  THE   MATERIALS. 


[Cn.  II . 


24.  Modulus    of    Elasticity    in    Compression. — The    stress- 
strain  line  being  curved  almost  from  the  beginning,  the  proper 


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Deformation  per  Unit  Length 

FIG.  2. — Compressive  S.tress-strain  Diagrams  of  Concrete. 

method   of   calculating  the   modulus  of  elasticity  needs   to  be 
considered.     Fig.  4  is  a  typical  stress-strain  diagram  for  com- 


24.] 


PROPERTIES  OF  CONCRETE. 


21 


pression  (somewhat  simplified),  B  and  C  being  points  where 
the  loads  have  been  removed  and  reapplied.  For  very  low 
stresses,  up  to  perhaps  300  to  400  lbs/in2  (a  low  working 
stress),  the  variation  of  the  curve  from  a  straight  line  is  so  small 


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FIG.  3. — Stress-strain  Diagram  under  Repeated  Loads. 

that  it  may  be  considered  as  straight,  and  an  average  straight  line 
may  be  drawn,  as  OT,  and  its  slope  taken  as  the  modulus  of  elas- 
ticity. This  line  may  be  considered  the  same  as  the  tangent 
at  the  origin.  For  higher  stresses,  reaching  to  a  point  along 


PROPERTIES  OF  THE  MATERIALS. 


[Cn.  II. 


the  curved  portion  such  as  point  B,  it  is  usual  to  deduct  the 
permanent  set  Oa  from  the  deformation  Ob  and  divide  the  stress 
by  the  remaining  elastic  deformation  ab.  This  gives  the  slope 
of  the  line  aB,  and  may  be  considered  to  represent  the  law  of 
elastic  deformation  for  stresses  within  the  limit  of  the  stress  bB 
after  the  first  few  applications  of  load.  A  modified  " elastic" 
curve,  OBf C",  can  thus  be  drawn  by  deducting  from  the  defor- 
mation for  each  load  the  subsequent  set,  giving  a  steeper  curve 
and  one  more  nearly  approaching  a  straight  line.  On  the  basis 


b  Deformation 
FIG.  4. 

of  this  " elastic"  curve  the  modulus  of  elasticity  for  stresses 
up  to  any  given  maximum  would  then  be  equal  to  that 
maximum  stress  divided  by  the  elastic  deformation  at  that 
stress. 

There  being  no  general  agreement  as  to  the  exact  definition  of 
the  word  "modulus"  for  such  materials  as  concrete,  the  method 
which  should  be  employed  in  calculating  its  value  should  depend 
upon  the  purpose  for  which  it  is  to  be  used.  The  principal 
use  of  the  modulus  of  elasticity  in  reinforced-concrete  design 
is  to  determine  the  relative  stresses  carried  by  the  concrete 
and  the  steel  in  compression  members,  and  to  find  the  neutral 
axis  in  beams.  After  the  neutral  axis  is  once  found  the  modulus 
does  not  enter  into  the  calculations. 


§24.]  PROPERTIES  OF  CONCRETE. 

Consider  the  action  in  the  case  of  a  column.  Assuming 
no  initial  stress  in  the  steel  or  concrete,  suppose  that  the 
column  is  loaded  so  as  to  cause  a  shortening  equal  to  06,  Fig. 
4.  The  stress  in  the  concrete  will  be  bB,  and  that  in  the  steel 
will  be  equal  to  the  deformation  Ob  multiplied  by  its  modulus 
of  elasticity.  Upon  removal  of  the  load  there  may  be  a  per- 
manent set  Oa,  which  means  that  there  is  some  residual  com- 
pression in  the  steel  (with  an  equal  amount  of  tension  in  the 
concrete).  A  second  application  of  the  load  will  cause  a 
deformation  ab,  but,  measuring  from  the  original  position,  the 
deformation  is  06,  and  this  again  fixes  the  stress  in  the  steel. 
Hence,  for  the  determination  of  the  relative  stresses  in  steel  and 
concrete  by  the  use  of  their  moduli  of  elasticity,  the -modulus 
for  the  concrete  should.be  the  ratio  of  Bb  to  06,  or  the  slope  of 
the  chord  OB.  That  is  to  say,  it  is  the  load  divided  by  the 
maximum  total  deformation  for  that  load.  This  ratio  will  be 
less  than  the  slope  of  the  line  OT,  or  of  the  elastic  line  aB. 

In  the  case  of  a  beam  the  stresses  in  the  concrete  at  any 
section  will  vary  from  zero  at  the  neutral  axis  to  the  value 
Bb,  for  example,  at  the  extreme  fibre.  At  intermediate  points 
the  stresses  follow  approximately  the  law  of  the  curve  OB. 
In  this  case  a  chord  OB  does  not  exactly  represent  the  facts, 
but  the  error  is  small,  and  it  is  the  best  line  to  use  if  the  rec- 
tilinear variation  of  stress  be  assumed.  If  a  curvilinear  law 
is  used,  then  the  modulus  is  supposed  to  be  the  slope  of  the 
tangent  at  the  origin.  In  neither  case  is  it  correct  to  use  the 
slope  of  the  line  aB. 

The  value  of  the  modulus  for  concrete  varies  greatly  as 
determined  by  different  experimenters  and  for  different  kinds 
of  concrete.  As  a  rule  the 'denser  and  older  the  concrete  the 
higher  the  modulus. 

Among  the  most  careful  experiments  are  those  by  Bach,* 
in  which  he  repeated  the  loads  at  each  increment  until  there 
was  practically  no  increase  of  set. 


*  Zeit.  V.  dt.  Ing.,  1895. 


24  PROPERTIES  OF  THE  MATERIALS. 

The  following  are  some  average  results: 


[Cn.  II. 


Kind  of  Concrete. 

Modulus  of  Elasticity,  lbs/in2. 

Based  on  Elastic  Deformation. 

Based  on  Total 
Deformation. 

At  114  lbs/in2. 

At  570  lbs/in2. 

At  570  lbs/in2. 

1'2|*5  (broken  stone)  

4,660,000 
3,170,000 
3,870,000 
3,000,000 

3,590,000 
2,520,000 
2,990,000 
2,240,000 

3,440,000 
2,200,000 
2,570,000 
2,110,000 

1-24-5  (gravel).  . 

1  :3:6  (broken  stone)  
1  •  3  •  6  (gravel) 

The  specimens  were  25  cm.  in  diameter  and  100  cm.  high 
and  were  from  three  to  four  months  old. 

The  average  values  of  the  modulus  obtained  in  the  Water- 
town  Arsenal  tests  mentioned  in  Art.  19  were  as  follows: 

TABLE  No.  2. 

MODULUS    OF   ELASTICITY    OF   CONCRETE. 
WATERTOWN  ARSENAL  TESTS,   1899. 


Mixture. 

Brand  of  Cement. 

Modulus  of  Elasticity  between  Loads  of  100 
and  600  lbs/in2,  Based  on  Elastic 
Deformation. 

7-10  Days. 

r  Month. 

3  Months. 

f 

1:2:4  \ 

\ 

•     •         { 
1:3:6  \ 

[ 

Savior 

1,667,000 
2,778,000 
1,000,000 
2,500,000 
2,500,000 

2,500,000 
3,125,000 
2,083,000 

2,778,000 

3,571,000 
.4,167,000 
4,167,000 
3,571,000 
2,778,000 

Atlas 

Alpha      

Germania  

Alsen  

Average 

2,089,000 

2,273,000 
1,667,000 

2,273',000 
1,667,000 

2,621,000 

2,778,000 
3,125,000 
2,083,000 
2,273,000 
2,273,000 

3,651,000 

4,167,000 
2,778,000 
3,571,000 
2,778,000 
2,778,000 

Say  lor. 

Atlas.  ...           

Alpha.  <  

Germania  

Alsen  . 

Average  

1,970,000 

2,506,000 

3,214,000 

These  results  were  calculated  by  using  the  total  deformation 
minus  the  set.  If  the  total  deformation  be  used  the  values 
would  be  reduced  in  most  cases  10  to  20  per  cent. 


§25.]  PROPERTIES  OF  CONCRETE.  25 

The  average  value  of  the  modulus  obtained  from  tests  made 
at  the  University  of  Wisconsin  of  30  cylinders  of  1:2:4  con- 
crete, 30  days  old,  6  in.  in  diameter  and  18  in.  high,  cal- 
culated at  a  stress  of  600  lbs/in2  and  using  the  total  defor- 
mation, was  2,560,000  lbs/in2.  The  average  compressive 
strength  was  1780  lbs/in2. 

Values  considerably  higher  than  most  of  those  already  quoted 
have  been  found  by  some  experimenters,  some  using  the  tan- 
gent at  the  origin  and  some  the  elastic  deformation  in  their 
calculations. 

Considering  the  various  results  obtained  and  the  signifi- 
cance of  total  deformation,  the  authors  would  suggest  for 
working  loads  a  minimum  value  of  2,000,000  and  a  maximum 
value  of  3,000,000,  depending  upon  richness  of  mixture  and 
age  of  concrete  for  which  the  calculations  are  made.  A  large 
number  of  tests  on  beams  noted  in  Chapter  IV,  in  which  the 
neutral  axis  was  carefully  determined,  gave  results  corresponding 
closely  to  a  value  of  2,000,000  for  the  modulus.  As  a  large 
variation  in  the  assumed  value  of  the  modulus  results  in  but 
small  variation  in  design,  no  great  accuracy  in  this  matter  is 
needed.  Where  ordinary  concrete  is  used  a  general  average 
value  of  2,500,000  is  sufficiently  close  for  all  practical  purposes. 

25.  Elastic  Limit. — As  stated  in  the  preceding  article,  con- 
crete shows  a  permanent  set  under  small  loads  so  that,  in  the 
usual  sense,  the  material  can  hardly  be  said  to  have  an  elastic 
limit.  There  appears  to  be,  however,  a  limit  to  the  stress 
which  can  be  repeated  indefinitely  without  continuing  to  add 
to  the  deformation,  and  this  limit  may  be  taken  as  the  elastic 
limit  for  practical  purposes.  From  experiments  by  Bach  and 
others,  this  limit  seems  to  be  from  one-half  to  two-thirds  the 
ultimate  strength.  In  repeated-load  experiments  on  neat 
cement  and  on  concrete  made  by  Professor  J.  L.  Van  Ornum  * 
it  has  been  shown  that  the  maximum  load  which  may  be 
repeated  an  indefinite  number  of  times  without  rupture  does 

*  Trans.  Am.  Soc.  C.  E.,  Vol.  LI,  p.  443.     Proc.  Am.  Soc.  C.  E.,  Dec.  1906. 


26  PROPERTIES  OF  THE  MATERIALS.  [Cn.  II. 

not  much  exceed  50%  of  the  ultimate  strength.*  These  results 
show  a  close  relation  to  those  obtained  by  Bach,  and  it  may 
therefore  be  concluded  that  the  limit  of  permanent  elasticity 
for  repeated  loads  is  from  50  to  60%  of  the  ultimate  strength, 

26.  Comparison  of  Stress-strain  Curve  with  the  Parabola.— 
As   the   parabola   is   often    used    in    theoretical    analyses    to 
represent  the  stress-strain  curve  it  will  be  useful  to  compare 
some  typical  curves  with  the  parabola.    The  form  of  parabola 
used  has  its  axis  vertical  and  its  vertex  at  the  point  of  the 
curve  representing  the  ultimate  strength.    In  Fig.  5  the  curves 
shown  in  Fig.  2  are  compared  with  parabolas  (shown  in  dotted 
lines) .    In  the  case  of  curves  C,  D,  E,  and  F  the  agreement  is 
very  close. 

27.  Stress-strain    Curve    for    Tension. — Comparatively  few 
tests  have  been  made  on  the  elasticity  of  concrete  in  tension. 
Professor  Hatt  found  the   average  value  of  the  modulus  for 
1:2:4  concrete,  30  days  old,  to  be  2,100,000  and  the  average 
total  elongation  at  rupture  VTOOO  part,  with  a  tensile  strength 
of  311  Ibs/in2.f     Later  tests  by  him  gave  for  the  modulus  the 
high  values  of  3-5,000,000,  which  were  about  the  same  as  the 
values  in  compression.     These  and  other  tests  indicate  that 
the  initial  modulus  in  tension  and  in  compression  are  about 
the  same,  and  as  the  working  limit  in  tension  is  very  low  they 
may  be  assumed  as  equal  without  material  error.    The  rela- 
tive strength  and  deformation  of  concrete  in  compression  and 
tension  is  illustrated  by  a  typical  curve  in  Fig.  6. 

28.  Coefficient  of  Expansion. — Experiments  by  Professor 
W.  D.  Pence  {  on  1 :2:4  concrete  gave  an  average  value  of  the 
coefficient  of  expansion  of   .0000055  per  degree  Fahrenheit, 
there  being  little  variation  among  the    several  tests.     Tests 
made  at  Columbia  University  on  1:3:6  concrete  gave  values 
of  about  .0000065.     Other  experiments  have  shown  somewhat 
h'gher  results.    A  value  of  .000006  may  be  assumed. 

*See  also  Art.  117. 

t  Proc.  Am.  Soc.  Test.  Mat.,  1902. 

J  Jour.  West  Soc.  Eng.,  Vol.  VI,  1901,  p.  549. 


§29.] 


PROPERTIES  OF  CONCRETE. 


27 


29.  Contraction  and  Expansion  in  Hardening. — Many  ex- 
periments have  been  made  relative  to  shrinkage  and  swelling 


'  .0002  .0004  .OC  ,06.0008  .0010  0  /  .0002 


.0001  .0006  .0008  .0010  .0012  .0014  .0016 


.0002..0004  .0006  .0008  .0010     0    ,0002  .0004  .0006  .0008  .0010  .0012 .0014  . 
Deformation  per  Unit  Length 

FIG.  5.— Stress-strain  Curves  Compared  with  Parabolas. 

of  cement-mortar  in  hardening.     In  general  the  results  show 
that  when  hardened  in  air  there  will  be  more  or  less 


28 


PROPERTIES  OF  THE  MATERIALS. 


[Cn.  II. 


but  when  hardened  in  water  there  is  likely  to  be  some  swelling, 
although  results  on  this  point  are  not  entirely  consistent. 
The  richer  the  mortar  (or  concrete)  the  greater  the  change  in 
dimensions.  Experiments  by  Consid£re  and  others  indicate 
that  1:3  plain  mortar  will  shrink  .05  to  .15%  when  hardened 


2400 


-2UOO 


-1800 


,0002      0 


100- 


.0002   .0004 


.0012 
Deformatioi 


^1 


FIG.  6. — Relative  strength  and  deformation  in  compression  and  tension. 

in  air  for  2  to  4  months,  and  neat  cement  from  two  to  three 
times  as  much.  Considere  found  the  shrinkage  of  a  1 : 3  mortar 
reinforced  with  5J%  of  steel  to  be  only  .01%,  or  one-fifth  the 
amount  his  tests  showed  on  plain  mortar.  The  few  tests 
available  show  that  the  shrinkage  of  concrete  is  less  than  that 


§31.] 


PROPERTIES  OF  CONCRETE. 


29 


of  mortar,  and  it  would  appear  that  the  shrinkage  should  be 
nearly  proportional  to  the  amount  of  cement  per  unit  volume, 
as  the  sand  and  stone  are  unaffected. 

30.  Weight  of  Concrete. — The  weight  of  concrete  of  the 
usual  proportions  will  vary  from  140  to  150  Ibs/f t3,  depending 
upon  the  degree  of  compactness  and  the  specific  gravity  of  the 
materials.    Variation  of  proportions  will  affect  the  weight  but 
little  if  the  proper  ratio  of  sand  and  stone  be  maintained,  but 
a  wet  concrete  when  dried  out  will  weigh  less  than  a  well-com- 
pacted concrete  containing  originally  less  water.     For  prac- 
tical purposes  an  average  value  of  145  lbs/ft3  may  be  taken. 
The  addition  of  reinforcing  steel  in  the  usual  proportions  will 
add  from  3  to  5  pounds,  so  that  the  weight  of  reinforced  con- 
crete may  be  taken  at  150  lbs/ft3. 

31.  Properties  of  Cinder  Concrete. — The  following  table  of 
results  indicates  fairly  well  the  strength  and  modulus  of  elas- 
ticity of  cinder  concrete.    The  age  of  the  specimens  varied 
from  30  to  100  days.    Cinder  concrete  will  weigh  from  110  to 
115  lbs/ft3. 

TABLE  No.  3. 

CRUSHING    STRENGTH    AND    MODULUS    OF   ELASTICITY    OF    CINDER 

CONCRETE. 

WATKRTOWN    ARSENAL   TESTS,    1898. 


Mixture. 

Average  Crushing  Strength, 
lbs./in.2. 

Average  Modulus  of  Elas- 

100  and  600  lbs./in.2. 

Cement. 

Sand. 

Cinders. 

One  Month. 

Three  Months. 

1 

1 

3 

1540 

2050 

2,540,000 

1 

2 

3 

1098 

1634 

1 

2 

4 

904 

1325 

1 

2 

5 

724 

1094 

1,040,000 

1 

3 

6 

529 

788 

30  PROPERTIES  OF  THE   MATERIALS.  [Cn.  II. 

REINFORCING   STEEL. 

32.  General  Requirements. — In   general,  reinforcing  steel 
must  be  of  such  form  and  size  as  to  be  readily  incorporated 
into  the  concrete  so  as  to  make  a  monolithic  structure.    To 
provide   the  necessary  bond  strength  and   to  distribute   the 
steel  where  needed  without  concentrating  the  stresses  on  the 
concrete  too  greatly,  requires  the  use  of  the  steel  in  compara- 
tively small  sections.     This  requirement,  as  well    as  that  of 
economy  and  convenience,  leads  to  the  use  of  the  steel  in  the 
form  of  rods  or  bars.     These  will  vary  in  size  from  about 
J  to  f  inch  for  light  floors  up  to  1J  to  2  inches  as  maximum 
sizes  for  heavy  beams  or  columns.     Under  certain  conditions 
a  riveted  skeleton  work  is  preferred  for  the  steel  reinforcement, 
but  this  is  usually  where  for  some  reason  it  is  desired  to  have 
the  steelwork  self-supporting  or  where  it  is  to  carry  an  unusu- 
ally large  proportion  of  the  load. 

33.  Forms  of  Bars. — Plain  round  rods  have  been  used  gen- 
erally in  Europe  for  many  years,  and  also  very  largely  in  this 
country,  adhesion  being  depended  upon  for  the  transmission 
of  stress.     Square  rods  show  about  the  same  adhesive  strength 
as  round  rods,  but  are  not  so  convenient  to  use  or  so  readily 
obtained.     Flat  bars  are  undesirable,  as  their  adhesion  to  the 
concrete  is  much  below  that  of  round  or  square  bars. 

Many  special  forms  of  bars  have  been  devised,  the  prin- 
cipal object  of  which  is  to  furnish  a  bond  with  the  concrete 
independent  of  adhesion, — a  "mechanical  bond"  as  it  is  usu- 
ally called.  Some  of  the  most  common  types  of  such  bars 
are  illustrated  in  Fig.  7.  Fig.  (a)  is  the  twisted  square  bar 
invented  by  Mr.  Ransome  and  called  by  his  name.  It  is 
usually  twisted  cold.  Figs.  (6),  (c),  and  (d)  illustrate  various 
well-known  types  of  deformed  bars  which  are  shaped  in  the 
rolling.  Fig.  (e)  illustrates  the  Kahn  bar,  formed  by  turning 
up  strips  sheared  from  the  thin  part  of  the  bar.  Many  other 
devices  are  employed  to  a  greater  or  less  extent  to  provide  a 
mechanical  bond,  and  a  great  variety  of  combinations  of  forms 


§33J 


PROPERTIES  OF  STEEL. 


31 


FIG.  7. — Deformed  Bars. 


32  PROPERTIES  OF  THE  MATERIALS.  [Cn.  II. 

are  used  in  the  construction  of  beams,  floors,  and  columns  as 
patented  "systems".  It  is  the  purpose  here  to  mention  only 
the  most  common  types  of  bar  element. 

34.  General  Quality  of  the  Steel.— Steel  used  in  reinforced 
work  is  not  usually  subjected  to  as  severe  treatment  as  that 
used  in  ordinary  structural  work.     Bars  must  be  capable  of 
being  bent  to  the  desired  form,  but  this  is  the  only  treatment 
to  which  the  ordinary  bars  are  subjected.     In  many  concrete 
structures  the  impact  effect  is  also  likely  to  be  less  than  in 
all-steel  structures;    consequently  it  is  considered  that  a  some- 
what less  ductile  material  may  safely  be  used,  but  to  what 
extent  these  considerations  should  permit  the  use  of  steel  of 
cheaper  grade  or  of  higher  elastic  limit  is  an  open  question  on 
which  there .  is  much  difference  of  opinion.    The  question  of 
elastic  limit  as  related  to  working  stresses  and  stresses  in  the 
concrete  is  discussed  in  Chapter  V. 

35.  Tensile  Strength. — Various  grades  of  steel  are  used  in 
reinforced  concrete  ranging  from  soft  to  quite  hard.     Classify- 
ing the  material  under  three  heads,  soft,  medium,  and  hard, 
the  elastic  limit  and  ultimate  strength  will  range  about  as 
follows : 

Soft.  Medium.  Hard. 

Elastic  limit,  lbs/in2 30-35,000       35-40,000      50-60,000 

Ultimate  strength,  lbs/in2..    50-60,000        60-70,000       80-100,000 

In  some  forms  of  rods  used  the  elastic  limit  is  artificially 
raised  by  cold  working. 

36.  Modulus  of  Elasticity. — The  modulus  of  elasticity  of  all 
grades  of  steel  is  very  nearly  the  same  and  will  be  taken  at 
30,000,000  lbs/in*. 

37.  Elastic  Elongation. — As  bearing  upon  deformations  the 
elongation  of  the  steel  at  its  elastic  limit  will  be  here  noted. 
Using  the  above  value  of  the  modulus  of  elasticity  the  elonga- 
tion per  unit  length  of  the  three  grades  of  steel  at  their  elastic 
limit  will  be  as  follows: 

Soft.., 0.0010-0.0012 

Medium. . ;... 0012-    .0013 

Hard.  .0017-    .0020 


§£9]  ADHESIVE  STRENGTH.  33 

38.  Coefficient  of  Expansion. — The  coefficient  of  expansion 
of  steel  may  be  taken  at  .0000065  per  1°  F. 


PROPERTIES   OF   CONCRETE   AND   STEEL   IN   COMBINATION. 

39.  Adhesion  of  Concrete  and  Reinforcing-bars.  —  The 

high  value  of  the  tangential  adhesion,  or  grip,  of  concrete  to 
steel  rods  embedded  therein  has  long  been  known  and  has 
been  utilized  in  the  placing  of  anchor-rods,  etc.  It  is  some- 
what remarkable,  however,  that  only  recently  has  this  property 
been  made  use  of  in  the  design  of  combination  structural  forms. 
Experience  has  shown  this  adhesion  to  be  sufficiently  reliable 
and  permanent  to  be  utilized  in  such  combination  structures, 
and  plain  smooth  bars  have  been  entirely  successful.  Bars 
of  irregular  section  in  which  adhesion  is  not  entirely  depended 
upon  for  the  bond  are  also  used  to  a  large  extent.  Some  form 
of  mechanical  bond  is  necessary  where  the  adhesion  area  is 
deficient,  and  some  engineers  consider  such  a  bond  desirable  in 
all  cases. 

Numerous  tests  have  been  made  by  various  experimenters 
to  determine  the  adhesion  between  concrete  and  plain  rods 
of  different  forms,  with  results  varying  from  about  200  to 
about  750  lbs/in2.  The  adhesive  strength  is  largely  frictional 
resistance  and  varies  greatly  with  the  roughness  of  the  bars. 
It  also  varies  with  the  quality  of  the  concrete  and  the  method 
of  conducting  the  test.  Usually  the  test  is  made  by  embedding 
the  rod  in  a  block  of  concrete  and  pulling  it  therefrom,  the  rod 
being  stressed  in  tension  and  the  concrete  in  compression. 
This  causes  a  maximum  of  elongation  in  the  steel  at  the  point 
where  it  enters  the  concrete,  while  the  concrete  is  subjected 
to  a  maximum  compression  at  this  same  point.  This  brings 
very  unequal  stresses  upon  the  adhering  surfaces,  tending  to 
a  progressive  separation  until  the  entire  rod  has  started  to  slip, 
after  which  friction  alone  holds  the  rod.  This  unequal  action 
is  greater  the  deeper  the  embedment.  If  the  rod  is  pushed  out, 
both  rod  and  concrete  are  compressed,  although  not  the  same 


34 


PROPERTIES  OF  THE   MATERIALS. 


[Cn.  II. 


amount  at  the  same  point.  Tests  made  in  this  way  should 
therefore  give  higher  results  than  where  the  rods  are  pulled 
out.  Experimental  results  accord  in  general  with  these  prin- 
ciples. 

In  a  beam,  conditions  are  more  favorable  than  in  tests 
conducted  by  either  method,  as  both  steel  and  concrete  are 
elongated,  thus  tending  to  distribute  the  stress  more  equally. 
Results  of  tests  made  in  the  usual  way  may  then  be  taken  as 
well  within  the  limit  of  what  may  be  expected  in  a  beam. 

The  following  table  contains  in  condensed  form  the  results 
of  the  most  important  tests  on  adhesion: 

TABLE  No.  4. 

ADHESION   TESTS. 


Steel  Rods. 

Authority. 

Kind  of 
Concrete. 

DepthEm- 
bedded, 
Inches. 

Adhesive 
Strength, 
lbs/in2. 

Kind. 

Size, 
Inches. 

Feret; 

1:2:4 

Plain  round 

0.8 

2f 

237 

Ciment  Arme,  p.  755. 

1:2:5 

tt        ti 

0.8 

2f 

190 

1:3:4* 

(  (        t  ( 

0.8 

2| 

237 

1:3:6 

1  1        t  ( 

0.8 

2} 

195 

Hatt; 

1:2:4 

Plain  round 

f 

6 

756 

Proc.  Am.  Soc.  Test 

1:2:4 

<  (         (  ( 

ft 

6 

636 

Mat.,  1902. 

Emerson; 

1:3 

Plain  round 

I 

6 

512 

Eng.  News,  Vol.  LI, 

1:3 

Plain  flat 

ixi 

6 

293 

1904,  p.  222. 

1:2:4 

Plain  square 

ixi 

10 

587 

1:3:6 

ii         it 

ixi 

10 

478 

Talbot; 

1:2:4 

Plain  round 

i  and  f 

6 

438 

Bull.  No.  8,  Univ.  of 

1:2:4 

(  t          tt 

i  and  f 

12 

409 

Itt.,  1906. 

1:3:5J 

i  c          (  ( 

J  and  f 

6 

364 

l:3:5i 

i  (          (  t 

2  and  f 

12 

388 

l:3:5i 

Cold  rolled 

shafting 

1  and  J 

6 

146 

1:3:5£ 

Mild  steel 

flat 

ftXli 

6 

125 

1:3:6 

Tool-steel 

round 

f 

6 

147 

Withey; 

1:2:4 

Plain  round 

ft  tof 

6 

401 

Bull.  Univ.  ofWis., 

1:2:4 

(i         1  1 

ft 

6 

504 

1937. 

$  40.]  ADHESIVE   STRENGTH.  35 

Tests  at  the  University  of  Wisconsin  gave  the  following  re- 
sults for  rods  of  different  sizes  and  for  6-inch  depth : 

A  in 329  lbs/in2  A  in 387  lbs/in2 

i  " 535      "  f  "  391      " 

|  "  ......454      "  |  "  327      " 

J  " 382      " 

These  results  show  little  or  no  effect  due  to  difference  in  size. 

Feret  found  an  increase  of  strength  at  age  of  two  years  of 
about  50%  over  that  at  three  months;  and  a  maximum  value 
for  quite  wet  concrete;  further,  that  a  small  amount  of  cor- 
rosion increased  the  value.  Bach  found  smaller  values  the 
greater  the  depth,  the  average  value  for  a  4-inch  minimum 
depth  in  1:4  gravel  concrete,  3  months  old,  being  470  lbs/in2. 
He  also  found  greater  values  when  the  rods  were  pushed  out 
than  when  pulled  out. 

Hatt  found  a  frictional  resistance,  after  starting,  of  50% 
to  70%  of  the  initial  strength,  and  Morsch  reports  such 
resistance  as  about  two-thirds  the  initial.  Talbot  determined 
the  frictional  resistance  in  a  large  number  of  tests,  finding  it 
to  range  quite  uniformly  from  about  55%  to  65%  of  the  initial 
or  bond  strength,  in  the  case  of  plain  round  or  flat  rods.  In  the 
case  of  cold-rolled  steel  the  friction  was  only  40%  of  the  bond 
strength. 

A  study  of  the  various  results  leads  to  the  conclusion  that 
for  ordinary  round  or  square  bars,  not  too  smooth,  the  adhesive 
strength  may  be  taken  at  from  300  to  400  lbs/in2,  with  a  fric- 
tional resistance  of  about  two-thirds  this  amount;  a  much 
smaller  value  must  be  taken  for  very  smooth  bars  and  also  for 
flat  bars. 

40.  Mechanical  Bond.— The  bond  strength  of  bars  with 
indented  surfaces  depends  upon  the  adhesive  resistance  and 
the  shearing  strength  of  the  concrete.  Under  heavy  stresses 
there  is  also  a  tendency  for  the  concrete  to  split,  owing  to  the 
tensile  stresses  developed.  The  bars  cannot  be  pulled  through 
the  concrete  without  shearing  off  an  area  equal  to  the  total 


36  PROPERTIES  OF  THE  MATERIALS.  [Cn  II. 

area  of  the  indented  portion  and  in  addition  overcoming  con- 
siderable friction  or  adhesion.  If  one-half  the  area  is  indented, 
the  bond  strength  can  then  be  placed  at  least  equal  to  one-half 
the  shearing  strength  (see  Art.  22),  or  about  one-fourth  the 
compressive  strength  of  the  material.  For  a  1:2:4  concrete 
this  would  equal  500  to  600  lbs/in2.  In  tests  of  such  bars 
failures  have  usually  occurred  by  the  splitting  of  the  specimen 
or  the  breaking  of  the  bar,  but  the  results  indicate  that  the 
actual  bond  strength  is  fully  equal  to  the  above  figures. 

An  important  phase  of  the  question  ( f  bond  relates  to  the 
amount  of  movement  which  may  take  place  under  loads  below 
tne  ultimate.  Tests  made  on  smooth,  twisted,  and  corrugated 
bars  and  reported  by  Mr.  T.  L.  Condron  *  showed  that  in  the 
case  of  the  plain  bars  and  most  of  the  twisted  bars  the  ultimate 
strength  was  very  nearly  reached  under  a  movement  of  Vioo 
inch.  In  the  case  of  the  corrugated  bars  the  load  causing 
ultimate  failure  (usually  due  to  the  splitting  of  the  concrete) 
was  in  some  cases  considerably  beyond  that  giving  a  movement 
of  Vioo  inch.  The  actual  stress  for  Vioo  inch  of  movement 
was  400-600  lbs/in2  for  the  corrugated  bars,  250-400  lbs/in2 
for  the  twisted  bars,  and  175-300  lbs/in2  for  the  smooth  bars. 
These  results  for  smooth  bars  are  notably  less  than  most  of 
those  given  in  Table  No.  4. 

41.  Ratio  of  Moduli  of  Elasticity,  ES:EC.— So  long  as 
the  adhesion  between  steel  and  concrete  is  unimpaired  the 
distortion  of  the  two  materials  will  be  equal.  Their  stresses 
will  then  be  proportional  to  the  moduli  of  the  elasticity  for  the 
load  in  question,  or  as  the  ratio  of  ES:EC.  Taking  E9  at 
30,000,000  and  Ec  at  from  2,000,000  to  3,000,000,  the  ratio 
varies  from  15  to  10.  In  practice  various  values  of  this  ratio 
are  used.  As  will  be  seen  in  Chapter  IV,  the  value  15  corre- 
sponds closely  to  actual  determinations  of  neutral  axes  in 
beams.  It  is  the  value  commonly  used  in  German  regulations 
and  is  also  used  in  the  building  laws  of  some  American  cities. 

*  Jour.  West.  Soc.  Eng.,  1907,  Vol.  XII,  p.  100. 


OF 


§42.]  EXTENSIBILITY  OF  CONCRETE.  37 

A  value  of  12  is  also  common  and  from  the  values  of  Ec  obtained 
from  compression  tests  as  given  in  Art.  24,  the  lower  value 
would  seem  to  be  more  nearly  correct. 

The  effect  of  a  variation  in  this  ratio  is  relatively  small. 
Thus  a  compression  member  containing  1%  of  steel,  and  with 
a  working  stress  in  the  concrete  of  fe,  will  have  a  strength  of 
P  =  fcA  +  fc(E8/Ec).QlA,  where  A  =  area  of  concrete.  IiE8/Ec= 
12,  P  =  /CA(1.12);  if  ES/EC  =  15,  P  =  /<4(1.15),  an  increase  of 
only  2.7%  for  a  change  in  ES/EC  of  25%. 

Equal  ratios  of  mpduli  may  be  assumed  for  both  tension 
and  compression. 

42.  Tensile  Strength  and  Elongation  of  Concrete  when 
Reinforced.  —  We  have  seen  that  plain  concrete  has  an  ulti- 
mate tensile  strength  of  about  200  lbs/in2  and  a  total  elon- 
gation of  perhaps  Vrooo  part,  corresponding  to  a  value  of 
1,400,000  for  Ec.  Steel  stretches  this  amount  under  a  stress 
of  30,000,000/7000  =  4300  lbs/in2.  Again,  the  safe  working 
tensile  stress  of  concrete  is  about  50  lbs/in2,  and  if  we  use  a 
value  of  E8/Ec=lbj  the  corresponding  stress  in  the  steel  will 
be  but  750  lbs/in2.  From  these  relations  it  is  evident  that 
in  reinforced  tension  members  we  must  either  use  very  low  and 
uneconomical  working  stresses  for  steel,  or  else  expect  the 
concrete  to  be  of  no  assistance  in  carrying  stress. 

In  studying  the  behavior  of  reinforced  concrete  under 
tension,  and  especially  when  constituting  the  tensile  side  of 
a  beam,  results  of  some  experiments  indicate  that  the  con- 
crete in  this  condition  elongates  more  before  final  rupture 
occurs  than  when  not  reinforced,  and  that  the  resistance  of 
the  concrete  is  nearly  constant  and  at  its  maximum  value 
for  some  time  previous  to  rupture.  The  first  to  announce 
this  principle  was  Considere,  whose  tests  indicated  that  the 
ultimate  stretch  of  reinforced  concrete  was  as  much  as  ten 
times  that  of  plain  concrete.  Kleinlogel,*  however,  was  unable 
to  check  these  results,  he  finding  an  elongation  of  practically 

*  Beton  u.  Eisen,  No.  2,  1904. 


38  PROPERTIES   OF  THE  MATERIALS.  [Cn.  II. 

the  same  amount  as  for  plain  concrete.  In  experiments  of 
this  sort  it  is  extremely  difficult  to  determine  just  when  the 
concrete  begins  to  crack.  The  steel  forces  it  to  elongate  prac- 
tically uniformly,  even  after  rupture  begins,  so  that  a  crack 
will  open  up  very  slowly  and  will  therefore  remain  almost 
invisible  for  some  time. 

In  some  experiments  made  at  the  University  of  Wisconsin 
in  1901-2  a  very  delicate  method  of  detecting  incipient  cracks 
was  accidentally  discovered.  It  was  found  that  beams  cured 
in  water  which  were  only  partially  dried  before  testing 
would,  when  tested,  show  very  fine  hair-cracks  at  an  early 
stage,  and  moreover,  by  watching  closely,  it  was  observed  that 
preceding  the  appearance  of  a  crack  there  would  appear  a 
dark  wet  line  across  the  beam.  Such  a  line  would  soon  be 
followed  by  a  very  fine  crack.  A  larger  series  of  tests  were  under- 
taken in  the  following  year  by  a  different  set  of  experimenters, 
who  observed  the  same  phenomenon.  Careful  measurements  of 
extension  showed  that  these  streaks  or  "water-marks ",  as 
they  were  named,  occurred  at  practically  the  same  deformation 
at  which  the  concrete  ruptured  when  not  reinforced.  Some 
of  the  results  are  given  in  Table  No.  5.*  The  beams  were  of 
1:2:4  mixture  by  weight  and  were  6"X6"  in  cross-section  by 
60  inches  span. 

That  these  water-marks  were  incipient  cracks  was  deter- 
mined by  sawing  out  a  strip  of  concrete  along  the  outer  part 
of  the  beam.  Fig.  8.  is  a  photograph  showing  the  results  of 
this  experiment.  Very  close  observation  also  in  many  cases 
showed  hair-like  cracks  appearing  very  soon  after  the  appear- 
ance of  the  water-marks. 

Comparing  the  observed  and  calculated  elongations  of  the 
reinforced  concrete  with  those  of  the  plain  concrete  at  rupture 
it  will  be  seen  that  the  initial  cracking  in  the  former  occurs  at 
an  elongation  practically  the  same  as  reached  by  the  latter  at 
rupture. 

*  Bulletin  No.  4,  Engineering  Series,  Univ.  of  Wis.,  1906. 


§42.] 


EXTENSIBILITY  OF  CONCRETE. 


TABLE  No.  5. 

TESTS  OF  BEAMS  SHOWING  EXTENSIBILITY  OF  CONCRETE. 


Proportionate  Extension. 

Compressive 

Tkr^A-L.  _  J 

No. 

Age. 

Metnoa 
of  Loading. 

At  First 
Water- 

At First 
Visible 

ot/rcngtn  of 
Cubes, 
Ibs/in2. 

mark. 

Crack. 

8 

3  months 

At  third  points 

.00011 

.00064 

4250 

10 

i  ( 

n 

.00024 

.00046 

2500 

22 

<( 

1  1 

.00025 

.00065 

2775 

26 

i  i 

«  « 

.00016 

.00056 

3000 

30 

(  { 

tt 

.00012 

.00064 

2600 

7 

1  .month 

At  center 

.00015 

.00036 

3500 

5 

t  < 

(  ( 

.00020 

.00031 

3500 

13 

i  e 

1  1 

.00009 

.00011 

2350 

23 

1  1 

1  1 

.00020 

.00060 

2500 

35 

(  ( 

1  1 

.00013 

.00053 

3150 

2* 

1  1 

n 

At  rupture 

.00013 

3000 

1* 

1  t 

<t 

.00010 

2500 

*  Nos.  2  and  1  were  plain  concrete  beams.  The  extensions  of  the  beams  loaded  at 
the  third  points  were  measured  by  extensometers;  those  of  the  center  loaded  beams  were 
calculated  from  deflections. 


FIG.  8. 


40  PROPERTIES  OF  THE  MATERIALS.  [Cn.  II. 

It  should  be  said  that  in  many  cases  the  first  "water-marks  " 
did  not  extend  entirely  across  the  beam  (the  beam  was  observed 
on  the  tension  face),  so  that  presumably  the  concrete  as  a 
whole  would  still  possess  some  tensile  strength.  It  would 
seem,  however,  that  these  experiments  on  a  large  number  of 
beams  show  quite  clearly  that  the  initial  failure  begins  at  the 
same  elongation  as  in  plain  concrete.  In  the  plain  concrete 
total  failure  ensues  at  once;  in  the  reinforced  concrete  rupture 
occurs  gradually,  and  many  small  cracks  may  develop  simul- 
taneously, so  that  the  total  elongation  at  final  rupture  will 
be  greater  than  in  the  plain  concrete.  In  other  words,  the 
steel  develops  the  full  extensibility  of  a  non-homogeneous 
material  that  otherwise  would  have  an  extension  corresponding 
to  the  weakest  section. 

The  presence  of  these  cracks  of  course  seriously  affects  the 
tensile  strength  of  the  concrete,  and  as  they  appear  at  an 
elongation  corresponding  to  a  stress  in  the  steel  of  5000  lbs/in2 
or  less,  it  would  seem  that  no  allowance  should  be  made  for 
the  tensile  resistance  of  the  concrete.* 

It  will  be  assumed  therefore  in  this  work  that  the  strength 
of  concrete  in  tension  may  be  considered  only  when  the  defor- 
mations are  well  within  the  ultimate  deformations  of  plain 
concrete.  Assuming  a  maximum  value  of  the  tensile  stress  of 
200  lbs/in2,  and  a  value  of  Ea/Ec  =  15,  the  corresponding  stress 
in  the  steel  would  be  200  X 15  =  3000  lbs/in2.  Usually  the  steel 
is  stressed  to  10,000  lbs/in2  or  more,  in  which  case  it  must  be 
assumed  that  the  concrete  is  more  or  less  ruptured  and  of  no 
value  as  a  tension  member. 

In  practical  design  the  most  important  question  which 
arises  is  how  far  a  concrete  may  be  cracked  without  exposing 
the  steel  to  corrosive  influences.  In  this  respect  it  seems  to 
the  authors  that  the  minute  cracks  which  appear  in  the  early 
stages  of  the  tests  can  have  very  little  influence. 

43.  Relative  Contraction  and  Expansion. — Temperature 
changes  affect  both  the  steel  and  the  concrete.  But  as  the 
coefficient  of  expansion  of  steel  is  .0000065  and  of  concrete 

*  These  conclusions  have  recently  been  further  substantiated  by  Bach. 
See  Zeit.  Ver.  Dt.  Ing.,  1907. 


§  43.J  CONTRACTION  AND  EXPANSION.  41 

.000006,  the  two  materials  will  be  but  slightly  stressed  because 
of  any  difference  in  their  rates  of  expansion. 

The  effect  of  shrinkage  in  hardening  is  more  serious.  As 
shown  in  Art.  29,  the  hardening  of  concrete  is  accompanied 
by  more  or  less  contraction  if  in  air,  or  expansion  (to  a  less 
degree)  if  in  water.  Concrete  which  is  unrestrained  either  by 
steel  reinforcement  or  by  exterior  attachment  will  shrink  or 
swell  proportionally  and  no  stresses  will  thereby  be  developed. 
If  restrained  by  reinforcing  material  only,  a  shrinkage  will 
develop  tensile  stresses  in  the  concrete  and  compressive  stresses 
in  the  steel. 

If  it  be  assumed  that  concrete  when  reinforced  tends  to 
shrink  the  same  amount  as  plain  concrete,  and  that  such  shrink- 
age is  prevented  only  so  far  as  the  stresses  developed  in  the 
steel  react  upon  the  concrete  and  cause  an  opposite  movement, 
then  it  will  be  found,  using  the  ordinary  values  of  the  modulus 
of  elasticity,  that  the  stresses  developed  in  both  the  concrete 
and  the  steel  will  be  large.  These  stresses  would  be  determined 
as  follows: 

Let  c=  coefficient  of  contraction  of  the  concrete; 

/c=unit  stress  in  concrete  (tensile); 

fa  =  unit  stress  in  steel  (compressive)  ; 

p  =  steel  ratio; 

n  =  Es/Ec. 

Then  the  net  contraction  per  unit  length  as  measured  by 
the  concrete  will  be  c-fc/Ec,  and  as  measured  by  the  steel  will 
be  fs/E8.  These  values  are  equal.  Also,  for  equilibrium,  fc  =  pfa. 
From  these  equations  we  get 


and 


r 

If,  for  example,  c=.0003,  Ec  =  2, 000,000,  n=15,  p=l%,  then 
/c  =  80  lbs/in2  tension  and  /s  =  8000  lbs/in2  compression.  If 
and  /8  =  7000  lbs/in2. 


42  PROPERTIES  OF  THE  MATERIALS.  [Cn.  II. 

It  is  doubtful  if  such  large  initial  stresses  actually  occur  in 
reinforced  concrete  due  to  shrinkage  in  hardening. 

The  experiments  of  Considere  on  the  actual  contraction 
of  reinforced  concrete,  already  quoted  in  Art.  29,  indicate 
that  the  deformation  is  less  than  the  above  theory  would  call 
for.  For  example,  the  observed  contraction  of  .01%  of  rein- 
forced mortar  would  call  for  a  stress  in  the  steel  of  only  about 
3000  lbs/in2,  and  in  the  concrete  of  only  30  to  60  lbs/in2.  In 
slowly  hardening,  with  the  steel  in  place,  there  is  probably  a 
gradual  adjustment  in  the  concrete  which  results  in  less  internal 
stress  than  the  experiments  on  plain  concrete  would  indicate. 
Where  the  structure  is  restrained  by  outside  supports  which  are 
relatively  more  rigid  than  the  reinforcing  steel,  the  stresses  in 
the  concrete  become  greater  and  may  easily  reach  the  limit 
of  the  tensile  strength,  thus  causing  cracks.  (For  further 
discussion  of  reinforcement  under  such  conditions,  see  Chapter 
V,  Art.  142.) 


CHAPTER  III. 

GENERAL  THEORY. 

44.  Kinds  of  Members. — Structural  members  are,  for  con- 
venience,  usually  divided  into  tension  members,   compression 
members,  and  beams,  according  as  the  forces  to  be  resisted 
produce  in  the  member  simple  tension,  simple  compression, 
or  simple  bending.    Bending  moment  is  often  accompanied 
by  tension  or  compression,  producing  what  are  called  combined 
stresses  of  bending  and  tension,  or  bending  and  compression. 
Since  reinforced  concrete  is  not  used  for  plain  tension  mem- 
bers the  analysis  will  be  confined  to  the  beam,  both  under 
plain  bending  and  under  combined  stresses,  and  to  the  com- 
pression member  or  column.    The  flat  slab  supported  on  four 
sides  will  be  considered  as  a  special  case  of  beam.    In  rein- 
forced-concrete  construction  the  beam  is  the  most  important 
element,  being  used  under  a  great  variety  of  conditions. 

45.  Relation  of  Stress  Intensities  in  Concrete  and  Steel. 
In  the  following  discussion  it  will  be  assumed  that  the  con- 
crete and  steel  adhere  perfectly  and  therefore  deform  equally. 
Nearly  all  reinforced-concrete  construction  is  dependent  upon 
this  equal  action  of  the  two  materials,  although  simple  adhesion 
is  not  always  entirely  depended  upon.    Many  types  of  deformed, 
or  roughened,  bars  are  used  so  as  to  give  the  steel  a  grip  inde- 
pendent of  the  adhesion,  and  in  other  cases  bars  are  bent  or 
anchored  at  the  ends,  but  in  all  cases  it  is  assumed  that  the 
materials    adhere    perfectly    and    therefore    deform    equally. 
Many  tests  show  that  under  proper  design  this  is  for  all  prac- 
tical purposes  true. 

43 


44  GENERAL  THEORY.  [Cn.  III. 

Since  the  modulus  of  elasticity  of  a  material  is  the  ratio 
of  stress  to  deformation,  it  follows  that  for  equal  deformations 
the  stresses  in  different  materials  will  be  as  their  moduli  of 
elasticity.  If 

/s=unit  stress  in  steel, 
/c=unit  stress  in  concrete, 
E8= modulus  of  elasticity  of  steel,  and 
Ec  =  modulus  of  elasticity  of  concrete, 
we  have  the  fixed  relation 

f./fc=E./Ec.  -1 (1) 

46.  Distribution  of  Stress  in  a  Homogeneous  Beam. — To 

assist  in  forming  correct  notions  of  the  action  of  steel  reinforce- 
ment in  a  concrete  beam,  it  will  be  desirable  to  consider,  at 
the  outset,  the  nature  of  the  stresses  due  to  bending  moment 
in  a  plain  concrete  or  homogeneous  beam  of  any  material. 
Considering  a  vertical  section  at  any  point  there  will  exist 
in  general  certain  normal  stresses  (tensile  and  compressive) 
and  certain  tangential  or  shearing  stresses.  A  knowledge  of 
these  stresses  on  a  vertical  section,  together  with  the  well-known 
principle  that  the  shearing  stress  at  any  point  is  of  equal  in- 
tensity vertically  and  horizontally,  is  sufficient  for  the  design- 
ing of  ordinary  beams. 

In  accordance  with  the  common  theory  of  flexure,  the 
normal  stress  on  a  vertical  section  varies  in  intensity  as  the 
distance  from  the  neutral  axis,  and  therefore  the  variation  is 
represented  by  the  ordinates  to  a  straight  line  as  in  Fig.  9. 

The  shearing-stress  intensity  is  a  maximum  at  the  neutral 
axis  and  is  zero  at  the  outer  fibres.  At  any  given  point  in  the 
section  it  is  given  by  the  equation 

v  =  VS/Ib, .     (1) 

in  which  V  denotes  the  entire  shear  at  the  section  containing 
the  point  under  consideration,  I  the  moment  of  inertia  of  the 
section  with  respect  to  the  neutral  axis,  b  the  breadth  of  the 
section  at  the  point,  and  S  the  statical  moment  of  the  part  of 


§46.] 


GENERAL  STRESS  DISTRIBUTION. 


45 


the  section  above  (or  below)  the  point  with  respect  to  the 
neutral  axis.  For  a  rectangular  beam  the  intensity  of  shear- 
varies  as  the  ordinates  to  a  parabola,  as  shown  in  Fig.  10,  the 
maximum  value  being  3/2  times  the  average,  or  equal  to 
3  V_ 
2'bd' 

If  the  stresses  on  inclined  planes  are  analyzed,  it  is  found 
,that  the  normal  and  shearing  stresses  will  not  be  the  same  as 
on  vertical  planes;  and,  furthermore,  that  wherever  shearing; 
stress  exists  on  a  vertical  plane  the  maximum  normal  stress; 


FIG.  9. 


FIG.  10. 


will  not  be  on  a  vertical  section,  but  on  an  inclined  one.  It  is 
proved  in  treatises  on  mechanics  that  if  /  represents  the  hori- 
zontal unit  tensile  stress  and  v  the  vertical  or  horizontal  unit 
shearing  stress  at  any  point  in  a  beam,  the  maximum  tensile 
stress  will  be  given  by  the  formula 

^i/Wi/a  +  tf8,      ......     (2) 

and  the  direction  of  this  maximum  tension  is  given  by  the 
formula  tan  26  =  2v/f,  where  6  is  the  angle  of  the  maximum 
tension  with  the  horizontal. 

A  study  of  these  formulas  shows  that  at  all  points  in  a  beam 
where  the  shear  is  zero,  the  direction  of  the  maximum  tension 
is  horizontal,,  as  at  points  of  maximum  bending  moment  and 
along  the  outer  fibres  of  the  beam.  Wherever  the  horizontal 
fibre  stress  is  zero  (at  the  neutral  surface  and  at  all  sections  of 
zero  bending  moment),  the  direction  of  the  maximum  tension 
is  inclined  45°  to  the  horizontal,  and  its  intensity  is  equal  to 
the  unit  shearing  stress  at  the  same  place.  Above  the  neutral 
axis  of  a  section  where  the  bending  moment  is  not  zero,  the 
inclination  of  the  maximum  tension  is  greater  than  45°,  becom- 


46 


GENERAL  THEORY. 


[CH.  III. 


ing  90°  at  the  upper  or  compressive  fibre.  Fig.  11  illustrates 
the  variation  in  normal  stress,  shearing  stress,  and  maximum 
tensile  stress  throughout  the  entire  depth  of  a  rectangular  beam. 
The  outer  normal  or  fibre  stress  is  assumed  at  200  lbs/in2,  and 
the  shearing  stress  at  the  neutral  axis  at  150  lbs/in2.  The 


Compression 


B  Tension  E 

FIG.  11. — Showing  Variation  of  Intensities  of  Normal  Stress,  Shear,  and 
Maximum  Tension. 

variation  in  the  fibre  stress  is  shown  by  the  straight  line  DE, 
and  that  in  the  shearing  stress  by  the  parabolic  curve  ACB. 
By  means  of  eq.  (2)  the  maximum  tensile  stresses  have  been 
computed;  these  are  represented  by  the  line  AHCJE. 

Fig.   12  illustrates  the  direction  of  the  maximum  tensile 


EIG.  12. — Lines  of  Maximum  Tension. 

stresses  in  a-  rectangular  beam.  The  exact  direction  at  any 
point  depends  upon  the  relation  between  shear  and  bending 
moment.  Lines  of  maximum  compression  would  run  at  right 
angles  to  the  lines  shown  and  lines  of  maximum  shear  at  angles 
of  45°  therewith. 

47.  Purpose  and  Arrangement  of  Steel  Reinforcement. — 
The  purpose  of  steel  reinforcement  is  to  carry  the  principal 
tensile  stresses,  the  concrete  being  depended  upon  for  the  com- 


§  48.]  COMMON  THEORY  OF  BEAMS.  47 

pressive  and  shearing  stresses,  its  resistance  to  such  stresses 
being  large.  If  no  steel  were  present  the  concrete  would  tend 
to  rupture  on  lines  perpendicular  to  the  direction  of  maximum 
tension,  as  shown  in  Fig.  12,  and  hence  we  may  conclude  that 
the  ideal  tension  reinforcement  would  require  the  steel  to  be 
distributed  in  the  beam  along  the  lines  of  maximum  tension.  \ 
At  the  centre  of  the  beam,  or  place  of  maximum  moment,  this 
direction  is  horizontal  for  the  entire  depth  of  the  beam,  and 
horizontal  rods  placed  near  the  lower  edge  of  the  beam  con- 
stitute proper  and  sufficient  reinforcement.  As  we  approach 
the  ends  of  the  beam,  where  the  shear  is  large,  the  intensity  of 
the  inclined  tensile  stresses  becomes  of  importance,  and  in  many 
cases  these  stresses  require  special  attention.  Horizontal  rods 
at  the  bottom  are  still  necessary,  but  do  not  entirely  reinforce 
the  concrete  against  tension,  so  that  special  consideration  must 
be  given  to  reinforcement  in  the  body  of  the  beam.  The 
arrangement  of  this  reinforcement  demands  careful  consid- 
eration. 

For  purposes  of  discussion,  the  subject  of  beams  will  first  be 
treated  with  reference  only  to  the  horizontal  reinforcement. 
The  inclined  tensile  stresses  will  be  considered  separately. 

48.  The  Common  Theory  of  Flexure  and  its  Modifi- 
cation for  Concrete. — The  common  theory  of  flexure  is  based  on 
two  main  assumptions,  namely,  (1)  a  plane  cross-section  of  an  un- 
loaded beam  will  still  be  plane  after  bending  (Navier's  hypoth- 
esis) ;  (2)  the  material  of  the  beam  obeys  Hooke's  law,  which 
is,  briefly  stated,  "stress  is  proportional  to  strain".  From  the 
first  assumption  it  follows  that — The  unit  deformations  of 
the  fibres  at  any  section  of  a  beam  are  proportional  to  their  dis- 
tances from  the  neutral  surface.  In  the  case  of  simple  bending 
(all  forces  at  right  angles  to  the  beam)  the  neutral  axis  lies  at 
the  centre  of  gravity  of  the  section;  in  the  case  of  bending 
combined  with  direct  tension  or  compression,  the  neutral  axis 
may  lie  in  the  section  or  be  merely  an  imaginary  line  without 
the  section.  From  the  second  assumption  it  follows  that — 
The  unit  stresses  in  the  fibres  at  any  section  of  a  beam  also  are 


48  GENERAL  THEORY.  [Cn.  III. 

proportional  to  the  distances  of  the  fibres  from  the  neutral  surface. 
This  may  be  called  the  linear  law  of  the  distribution  of  stress. 

The  linear  law  is  the  basis  of  all  practical  flexure  formulas 
excepting  some  for  reinforced-concrete  beams.  It  is  true  that 
wrought  iron  and  steel  are  the  only  important  structural  mate- 
rials which  closely  obey  Hooke;s  law,  and  they  only  within 
their  elastic  limits.  But  under  working  conditions  these  mate- 
rials are  not  stressed  beyond  these  limits,  and  so  the  formulas 
ordinarily  hold.  Timber,  stone,  and  cast  iron  can  hardly  be 
said  to  obey  Hooke's  law,  yet  for  working  conditions  the  com- 
mon flexure  formulas  for  these  materials  are  roughly  correct 
and  they  are  in  general  use. 

In  the  case  of  those  materials  which  do  not  obey  Hooke's 
law,  as  concrete,  and  for  all  materials  when  stressed  beyond 
their  elastic  limit,  the  common  theory  does  not  strictly  apply. 
An  exact  analysis  requires  the  use-  of  the  actual  tension  and 
compression  stress-strain  diagrams  for  the  materials  up  to  the 
limit  of  the  actual  stresses  involved.  It  will  be  assumed  still 
that  plane  sections  remain  plane  during  bending  so  that  defor- 
mations will  be  proportional  to  the  distances  of  the  fibres  from 
the  neutral  surface.  The  experiments  by  Talbot,*  though  not 
conclusive,  bear  out  this  assumption  in  the  more  important 
case  of  reinforced  beams.  Experiments  by  Schule,f  however, 
seem  to  show  that  original  plane  sections  do  not  remain  plane. 
Nevertheless  Navier's  hypothesis  will  probably  remain  a  basis 
of  flexure  formulas  for  reinforced-concrete  beams. 

The  variation  of  the  normal  stress  on  the  cross-section  can 
then  be  represented  graphically  in  the  following  manner:  Let 
Fig.  13a  be  the  stress-strain  diagram,  compression  above  the 
x  axis  and  tension  below,  for  the  material  in  question  as 
determined  by  direct  compression  and  tension  tests.  These 
curves  are  plotted  with  unit  stresses  as  abscissas  and  unit 
strains  as  ordinates.  Let  Fig.  136  represent  the  beam,  cut 

*  Univ.  of  111.  Bull.,  Vol.  II,  No.  1,  p.  28. 

f  Mitteilungen  der  Materialpriifungs-Anstalt  am  Po'ytedinikum  in  Zurich, 
Vol.  X  (1906),  p.  40. 


§49.] 


COMMON  THEORY  OF  BEAMS. 


49 


on  section  AB  where  the  stresses  are  to  be  investigated. 
The  neutral  axis  is  at  N.  Since  the  deformations  of  the  fibres 
are  proportional  to  the  distances  of  the  fibres  from  the  neutral 
axis,  these  distances  themselves,  Nl,  N2,  N3,  etc.,  will  represent 
to  some  scale  the  deformations.  If  the  unit  deformation  at 
point  1  is  then  represented  by  Nl  the  corresponding  stress  can 
be  determined  from  the  diagram  of  Fig.  13a,  using  the  proper 
scale  in  both  cases.  Lay  off  the  distance  la  to  represent 
that  stress.  Proceeding  similarly  for  all  points  and  connecting, 
we  have  the  stress  curve  A'NB',  which  is  nothing  more  than 


FIG.  13. 


a  portion  of  the  diagram  of  Fig.  13a  plotted  to  a  different 
scale. 

49.  Resisting  Moment  and  Inefficiency  of  Concrete 
Beams.  —  For  use  in  the  following  and  other  discussions  on 
flexure  three  important  principles  from  the  mechanics  of  beams 
are  now  recalled: 

(1)  For  beams  rectangular  in  section,   the  average  unit 
tensile  and  compressive  fibre  stresses  on  any  cross-section  are 
represented  by  the  average  abscissas  in  the  tensile  and  com- 
pressive parts  of  the  stress  diagram,  NBBf  and  NAA',  respect- 
ively (Fig.  136).      Also  the  whole  tension  T  and  whole  com- 
pression C  on  the  cross-section  are  proportional  to  the  areas 
NBB'  and  NA A'-,    hence,  according  to  some  scale,  the  areas 
represent  T  and  C  respectively. 

(2)  The  resultant  tension  T  and  resultant  compression  C 
act  through  the  centroids  of  the  tensile  and  compressive  areas 
in  the  stress  diagram. 

(3)  When  all  the  forces  (loads  and  reactions)  applied  to 


50  GENERAL  THEORY.  [Cn.  III. 

the  beam  act  at  right  angles  to  it,  then  the  resultant  tension  T 
equals  the  resultant  compression  C;  hence  the  two  stresses 
constitute  a  couple — "the  resisting  couple". 

Fig.  14  is  a  stress-strain  diagram  of  a  gravel  concrete  for 
both  tension  and  compression.  For  any  section  of  a  beam 

made  of  this  concrete,  the  stress 

/ 

diagram  is  a  certain  part  of 
the  stress-strain  diagram,  the 
exact  part  depending  on  the 
loading.  Suppose  that  the  loads 
produce  in  the  lower  fibre  at 
the  section  in  question  a  unit 
stress  represented  by  W  say, 
then  T  is  represented  by  NW 
and  C  by  an  area  Naaf  deter- 
mined from  the  principle  that 
FlG>  14>  it  must  equal  the  area  NW. 

Hence   the  stress   diagram   is 

aa'Nb'b,  and  the  unit  stress  on  the  upper  fibre  is  represented 
by  ad.  Furthermore,  ab  represents  the  depth  of  the  beam, 
and  N  the  position  of  the  neutral  axis.  Likewise,  when  the 
unit  stress  on  the  lower  fibre  is  BE'  (the  ultimate  tensile  strength) 
and  the  beam  is  on  the  point  of  failing,  T  is  represented  by 
the  area  NBB',  and  C  by  the  equal  area  NA A' ;  hence  the  stress 
diagram  for  the  failure  stage  is  AA'NB'B,  and  the  unit  stress 
on  the  upper  fibre  is  A  A' . 

50.  Resisting  Moment. — The  resisting  moment  of  a  section 
is  the  moment  of  the  resisting  couple  which  acts  at  that 
section.  Its  value  is  the  product  of  the  tension  (or  com- 
pression) and  the  distance  between  the  centroids  of  these 
stresses.  For  example,  at  the  failure  stage  of  the  beam 
above  referred  to  the  average  unit  tensile  stress  scales 
128  lbs/in2,  and  NB  =  0.6ZB  =  0.6d,  d  denoting  depth  of 
beam.  Hence  if  b  denotes  the  breadth  of  the  section, 

C=T=  128X0.6^X6-  76.8k/. 


§51.]  COMMON  THEORY  OF  BEAMS.  51 

The  vertical  distance  between  the  centroids  of  the  shaded 
parts  (NAAr  and  NBB')  of  the  diagram  is  0.64A#;  hence 
the  arm  of  the  resisting  couple  is  0.64d,  and  the  computed 
ultimate  resisting  moment  of .  a  beam  made  of  the  concrete 
under  consideration  is  76.86dx0.64d=49.26d2  in-lbs.,  6  and  d 
to  be  expressed  in  inches. 

Partly  to  test  the  correctness  of  the  theory  of  flexure  of 
concrete  beams,  Professor  Morsch  *  made  three  beams  15  X  20  cm. 
in  section  and  several  tension  and  compression  specimens  of 
the  same  mix  of  concrete.  From  tests  on  the  specimens  he 
obtained  a  stress-strain  diagram  from  which  he  computed 
the  probable  resisting  moment  of  the  beams  to  be  3A5bd2  = 
3.45 X 15 X202  =  21,700  kg-cm.  The  average  of  the  actual 
resisting  moments  of  the  beams  (determined  from  tests  to 
destruction)  was  22,100  kg-cm.;  an  agreement  to  be  regarded 
as  highly  satisfactory. 

The  working  resisting  moment  of  a  rectangular  beam  can 
be  computed  from  the  stress-strain  diagram  for  the  material 
in  this  same  manner.  Fortunately,  engineers  are  not  called 
upon  to  compute  resisting  moments  by  this  method.  It  is 
here  set  forth  principally  as  a  means  of  introducing  important 
ideas  bearing  on  reinforced-concrete  beams. 

51.  Inefficiency  of  Concrete  Beams. — When  a  beam  of  the 
concrete  above  referred  to  is  loaded  to  the  breaking  point, 
the  greatest  unit  compressive  stress  in  the  beam  is  the  stress 
AA'j  which  is  in  this  case  about  375  lbs/in2.  This  is  very  low 
compared  to  the  ultimate  compressive  strength  (2500  lbs/in 2), 
and  the  difference  indicates  a  wasteful  use  of  concrete. 

The  unshaded  portion  of  the  stress-strain  diagram  (Fig.  14) 
is  also  significant  in  this  connection,  for  it  indicates  the  unused 
compressive  strength  of  the  concrete  above  the  neutral  surface 
when  the  tensile  strength  of  that  below  is  fully  developed 
and  the  beam  is  about  to  fail. 

Another  way  to  express  the  inefficiency  of  a  concrete  beam 

*  Der  Eisenbatonbau. 


52  GENERAL  THEORY.  [Cn.  III. 

is  to  compare  its  ultimate  resisting  moment  with  that  which 
it  would  have  if  the  tensile  strength  and  elastic  properties 
were  the  same  as  the  compressive.  On  this  supposition  the 
tensile  stress-strain  diagram  would  be  like  the  compressive; 
and  for  the  concrete  of  Fig.  14,  the  ultimate  C  and  T  are  rep- 
resented by  the  area  NYY',  and  the  arm  of  the  resisting  couple 
by  twice  the  vertical  distance  of  the  centroid  of  the  area  NYY' 
above  N.  Actual  measurement  of  the  area  and  distance  gives 
0=775bd  and  arm=0.64d;  hence  the  ideal  ultimate  resisting 
moment  is  7756dx0.64d=4966d2  as  against  49.26d2,  the  actual 
value. 

To  supply  the  deficiency  in  tensile  strength  of  concrete  is 
the  main  purpose  of  steel  reinforcement.  A  comparatively 
small  amount  of  steel  (rods  or  bars  whose  combined  sectional 
area  is  from  1  to  2  per  cent  of  the  total  sectional  area  of  the 
beam)  properly  embedded  will  so  strengthen  the  tensile  side 
of  the  beam  that  the  great  strength  of  the  compressive  side 
can  be  utilized.  The  exact  amount  of  steel  required  in  any  case 
depends  on  the  elastic  properties  of  the  concrete  and  steel. 

52.  Varieties  of  Flexure  Formulas. — Many  formulas  have 
been  proposed  for  the  strength  of  reinforced-concrete  beams. 
The  differences  among  them  arise  principally  from  three  sources, 
namely:  (1)  The  method  of  applying  the  factor  of  safety, 
(2)  the  law  of  distribution  of  the  compressive  fibre  stress  in 
the  concrete,  and  (3)  the  value  of  the  tensile  fibre  stress  in 
the  concrete.  In  regard  to : 

(1)  Two  views  are  held  as  to  the  proper  method  of  applying 
the  factor  of  safety.  For  example,  to  ascertain  the  safe  load 
for  a  given  beam,  some  engineers  assume  working  strengths  for 
the  concrete  and  steel,  with  which,  by  means  of  a  suitable 
flexure  formula,  they  compute  the  safe  load  directly;  other 
engineers  compute  the  breaking  load  of  the  beam  by  a  suitable 
formula  and  then,  with  reference  to  this  load,  they  decide 
upon  the  safe  load.  (The  pros  and  cons  of  these  two  methods 
are  discussed  in  Art.  118.)  Formulas  for  working  conditions 
(for  use  in  the  first  method)  are  explained  in  Arts.  54-9;  those 


§52.] 


COMMON  THEORY  OF  BEAMS. 


53 


for  ultimate  conditions   (for  use  in  the  second  method)   in 
Arts.  60-4;   and  those  for  both  conditions  in  Arts.  65-70. 

(2)  As  already  explained  in  Art.  48,  the  distribution  of  the 
compressive  fibre  stress  can  be  represented  by  a  portion  of 
the  stress-strain  diagram  for  the  concrete.  As  shown  in 
Art.  23,  the  stress-strain  curve  for  concrete  up  to  and  even 
beyond  working  stresses  is  nearly  straight,  and  the  most  widely 
used  flexure  formulas  for  working  conditions  are  based  on  the 
assumption  that  the  stress-strain  curve  is  practically  straight 
up  to  working  stresses.  Formulas  of  Arts.  54-9  and  all  other 
flexure  formulas  of  this  book  (except  those  of  Arts.  60-70) 
are  based  on  this  assumption.  When  the  curvature  of  the 
stress-strain  curve  has  been  taken  into  account,  it  has  gen- 
erally been  assumed  to  be  an  arc  of  a  parabola,  the  vertex 


FIG.  15. — Distribution  of  Fibre  Stress  in  Concrete  According  to  Various 

Assumptions. 

being  taken,  by  some,  at  the  end  of  curve  (the  ultimate  strength 
end)  and,  by  others,  beyond  that  point.  The  formulas  of  Arts. 
60-70  are  based  on  a  parabolic  stress-strain  curve,  the  vertex 
being  at  the  end. 

(3)  As  explained  hi  Art.  42,  when  a  reinforced-concrete 
beam  p  being  loaded,  the  concrete  adjoining  the  steel  fails 
(cracks)  probably  always  before  the  stress  in  the  steel  reaches 
5000  lbs/in2,  and  when  the  stress  reaches  working  values  the 
cracks  will  have  extended  well-nigh  to  the  neutral  surface. 
The  amount  of  tension  remaining  in  the  concrete  at  the  section 


54  GENERAL  THEORY.  H.  III. 

of  the  crack  is  comparatively  small,  and  this  tension  being 
near  the  neutral  surface,  the  resisting  moment  due  to  it  is 
also  small  compared  to  that  due  to  the  tension  in  the  steel. 
In  a  certain  formula  for  ultimate  resisting  moment  in  which 
this  residual  tension  in  the  concrete  is  allowed  for,  the  value 
of  the  term  expressing  the  contribution  of  this  tension  is  less 
than  }  per  cent  of  the  total  moment.  It  is  the  almost  universal 
practice  to  neglect  this  tension  entirely  in  flexure  formulas; 
this  practice  is  followed  in  this  book. 

An  idea  of  the  variety  of  flexure  formulas  proposed  can  be 
gained  from  Fig.  15,  which  shows  nine  distributions  of  fibre 
stress  in  the  concrete  according  to  as  many  different  formulas. 
53.  Notation. — Fuller  explanations  of  some  of  these  sym- 
bols are  given  in  subsequent  articles  where  the  formulas  are 
derived;  see  also  Fig.  16. 

*  fs  denotes  unit  fibre  stress  in  steel; 

"-/c       "         ll      "       "     "  concrete  at  its  compressive  face; 
e8       lt        "    elongation  of  the  steel  due  to  fa- 
ec       "         lt    shortening  of  the  concrete  due  to  /c; 
E8       "      modulus  of  elasticity  of  the  steel; 
-  Ec       "  "      "  the  concrete  in  compression; 

<     n       "      ratio  ES/EC\   . 

T        ll      total  tension  in  steel  at  a  section  of  the  beam; 
C       ' '      total  compression  in  concrete  at  a  section  of  the 

beam; 

M8       "      resisting  moment  as  determined  by  steel; 
M c       ' '       resisting  moment  as  determined  by  concrete ; 
^  M       "      bending  moment  or  resisting  moment  in  general; 
b       "       breadth  of  a  rectangular  beam; 
d       "      distance  from  the  compressive  face  to  the  plane 

of  the  steel; 
k       "      ratio  of  the  depth  of  the  neutral  axis  of  a  section 

below  the  top  to  d', 
I*  j       "      ratio  of  the  arm  of  the  resisting  couple  to  d; 

A       tf       area  of  cross-section  of  steel; 
,/  p        "       steel  ratio,  A/bd. 


§55.] 


FORMULAS  FOR  WORKING   LOADS. 


55 


54-  Flexure  Formulas  for  Working  Loads  Based  on 
Linear  Variation  of  the  Compression  and  Neglecting  Ten- 
sion in  the  Concrete. — The  loads  being  working  loads,  the  unit 
stress  in  the  steel  is  within  the  elastic  limit,  and  the  unit  stresses 
in  the  concrete  vary  as  the  ordinates  to  the  compressive  stress- 
strain  curve  for  concrete  up  to  working  stresses.  This  curve 
is  nearly  straight;  it  will  be  assumed  straight  to  simplify  the 
formulas.  The  resulting  errors  are  small,  as  is  explained  in 
Art.  70. 

55.  Neutral  Axis  and  Arm  of  Resisting  Couple. — It  follows 
from  the  assumption  of  plane  sections  that  the  unit  deformations 
of  the  fibres  vary  as  their  dis- 
tances from  the  neutral  axis; 
hence,  e8/ec=(d-kd)/kd  (see 
Fig.  16).  Also  es=fs/E8  and 
ec=fc/Ec]  hence,  introducing 
the  abbreviation  n} 

f8     d-kd    l-k 
k    ' 


fr 


nfc       kd 


(a) 


Jd 


FIG.  16. 


When  the  loads  and  reactions 
are  vertical  —  beam  horizontal 
—the  total  tension  and  compression  on  the  section  are  equal,  i.e., 

.     .     (b) 


Eliminating  /,//«,  between  equations  (a)  and  (b)  and  introduc- 
ing the  abbreviation  p  gives  2pn(l-  k)  =  k2;  this  if  solved  for  k 
gives 

pn  ......     (1) 


This  formula  shows  that  the  neutral  axes  of  all  beams  of  a 
given  concrete  and  of  a  given  percentage  of  reinforcement  are 
at  the  same  proportionate  depth,  k,  for  all  working  loads.  The 
lower  group  of  curves  in  Fig.  17  gives  k  for  different  values  of 
p  and  n;  thus  for  p  =0.015  (percentage  of  steel  =  1.5)  and 
n  =  15;  &=0.48.  The  curves  show  that  k  increases  as  p  or  n 
increases. 


\ 


56  GENERAL  THEORY.  [Cn.  Ill 

The  distance  of  the  centroid  of  the  compressive  stress  from 
the  compressive  face  of  the  beam  is  %kd;  therefore  the  arm  of 
the  resisting  couple,  TC,  is  given  by 

jd  =  d-\kd,     or     j=l  -p  .....     (2) 

As  k  increases,  /  decreases,  but  not  in  the  same  ratio.  Fig.  17 
shows  how  j  changes  with  p  for  four  different  values  of  n.  It 
should  be  noticed  that  /  does  not  vary  much  with  p,  and  that 
for  7i  =  15  and  p  between  0.75  and  1.0%  —  common  values— 
the  average  value  of  j  is  about  J. 

56.  Resisting  Moment  for  Given  Working  Stresses  /„  and  fc  — 
If  the  beam  is  under-reinforced,  its  resisting  moment  depends 
on  the  steel  and  its  value  then  is 


(3) 


If  over-reinforced,  the  resisting  moment  depends  on  the  concrete 
and  its  value  then  is 

(4) 


To  find  the  resisting  moment  in  a  given  case,  these  values  of 
M  must  be  compared,  and  the  lesser  one  taken;  but  it  may  be 
noticed  that  a  comparison  of  the  quantities  }8p  and  %fck  is 
sufficient  to  determine  which  of  the  values  is  the  lesser. 

For  approximate  computations  one  may  use    the  average 
values   /=!   and   &  =  f;    then  formulas    (3)   and    (4)   become 
respectively 

M,=i,A-ld,    .......     (3)' 

Me-/..J6(P  ........     (4)' 

57.  Unit  Fibre  Stresses  for  a  Given  Bending  Moment.— 
Formulas  for  these  may  be  obtained  from  equations  (3)  and  (4) 
by  solving  them  for  fs  and  fc  respectively;  M  will  denote  bend- 
ing moment.  Or,  one  may  reason  as  follows  :  Since  the  resisting 
moment  is  Tjd, 

M  T 

T=-r-d    and     /«--£;   ......     (5) 


J  57.] 


FORMULAS  FOR  WORKING   LOADS. 


57 


-n^-W- 

£$ 

JR 


15S 


-30- 


i 


«  s 
§^ 


Percentage  of  Steel 
It 


FIG.  17. 


58  GENERAL  THEORY.  [Cn.  III. 

also,  since  fc  equals  twice  the  average  unit  compressive  stress 
on  the  section,  and  C=  T, 

2T  2 


Approximating  as  before,  i.e.,  using  average  values  /--=  J  and 
=  f,  formulas  (5)  and  (6)  become  respectively 

M  T 

T=-    and    /.--,  .  ;..   ....     (5)' 


2T 

and  «- 


58.  Determination  of  Amount  of  Steel  and  Cross-section 
of  Beam  for  a  Given  Bending  Moment. — If  k  be  eliminated  be- 
tween equations  (a)  and  (6),  the  following  formula  for  steel 
ratio  results : 

1/2  (7) 

":.     '     M+1) 

It  shows  that  for  given  concrete  and  ratio  of  working  stresses, 
p  has  the  same  value  for  all  sizes  of  beams.  Fig.  18  gives 
graphically  the  proper  values  of  p  for  different  ratios  fs/fc 
and  four  different  values  of  n. 

If  a  value  of  p  less  than  that  given  by  (7)  is  adopted  then 
the  cross-section,  or  bd2  rather,  should  be  determined  from  the 
first  of  equations  (8),  if  greater,  from  the  second.  (These  are 
(3)  and  (4)  solved  for  bd2  respectively.) 

M           2       M 
M2  =  J^'     M*  =  ^i       (8) 


Values  of  k  and  j  can  be  obtained  from  (1)  and  (2)  or  Fig.  17; 
then  inserting  an  assumed  value  of  6,  d  can  be  obtained  by 
direct  solution  of  the  formula. 


FORMULAS  FOR  WORKING  LOADS. 


59 


fjyimate  Design. — To  determine  the  percentage  of 
steel,  use  (6)'  in  this  form,  p  =  i% /<•//«•  If  a  smaller  percentage 
than  this  is  decided  upon,  use  the  first  of  equations  (8)'  to 
determine  b  and  d\  and  if  a  larger  then  the  second  one. 


bd2 


M 


M 

— 

6/C 


(8)' 


59.  Diagrams    and   Examples. — Some   numerical    examples 
illustrating  the  preceding  principles  will  now  be  given,  and 


FIG.  18. 

then  some  diagrams  will  be  explained  by  means  of  which  com- 
putations in  such  examples  can  be  wholly  avoided  or  nearly  so. 

(1)  A  concrete  beam  is  10X16  inches  in  cross-section  and  the  tension 
reinforcement  consists  of  four  f  inch  steel  rods,  their  centres  being  two 
inches  above  the  lower  face  of  the  beam.  The  working  stress  of  the  con- 
crete being  600  Ibs/iri2  and  that  of  the  steel  15,000,  what  is*  the  safe 
resisting  moment  of  the  beam? 

Solutions.  The  cross-section  of  one  steel  rod  is  0.442  in2,  hence 
A  =  1.768;  and  as  6  =  10  and  e?=14,  p=  1.768/140  =  0.0126.  There- 
fore, n  being  taken  as  15,  from  (1)  fc  =  0.453;  also  from  (2)  ;=0.849. 
As  determined  by  the  steel,  the  resisting  moment  is  (see  eq.  3) 

3/«=  15,000 X  1.768 X0.849 X 14  =  315,000  in-lbs. 


60  GENERAL  THEORY.  [Cn.  III. 

As  determined  by  the  concrete,  the  resisting  moment  is  (see  eq.  4) 
Me  =  300  X  10  X  0.453  X  14  X  0.849  X  14  =  227,000  in-lbs. 

The  safe  resisting  moment  is  the  latter  value. 

The  approximate  formulas,  (3)'  and  (4)',  give  respectively 

M8=  15,000  X  1  .768  X  £  X  14  =  325,000 
and  Mc  =  600  X  i  X  10  X  142  =  !96,000  in-lbs. 

The  approximate  formula  relating  to  the  steel  always  gives  a  closer 
result  than  the  other. 

(2)  Suppose  that  the  beam  of  the  preceding  example  is  19  in.  deep 
and  is  subjected  to  a  bending  moment  of  350,000  in-lbs.      Compute  the 
greatest  unit  stresses  in  the  steel  and  concrete. 

Solutions.  The  steel  ratio  is  1.768/170  =  0.0104;  and  with  n  =  15, 
eq.  (1)  gives  A:  =  0.424;  and  eq.  (3)  gives  /  =  0.859.  Therefore 
T  =  350,000/0.859  X  17  -  24,000  Ibs.,  and  /«  =  24,000/1  .768  =  13,600  lbs/in2. 
Also  see-eq.  (6),  /c  =  48,000/0.424x10x17  =  665  lbs/in2. 

The  approximate  formulas  (5)'  and  (6/  give  respectively 

/.  =  13,500     and    /c  =  750  lbs/in2. 

Again,  of  the  approximate  formulas,  the  one  relating  to  the  steel  gives 
the  closer  result. 

(3)  A  beam  is  to  be  figured  to  withstand  a  bending    moment  of 
135,000  in-lbs.,  the  working  strength  of  the  concrete  and    steel  being 
taken  at  700  and  12,000  lbs/in2  respectively. 

Solutions.  For  n  =  15,  eq.  (7)  gives  p  =  0.0136.  With  this  value  of 
p,  eq.  (1)  gives  k  =  0.462,  and  hence  /  =  0.846.  Eq.  (8)  now  gives 


12,000X0.0136X0.846 

Many  different  values  of  b  and  d  will  satisfy  the  last  equation.     If  b  is 
taken  as  7  in.,  then 


,     or       =n. 
Finally  A  =0.0136(7X12)  =  1.14  in2. 

The  approximate  formula  6'  gives  for  a  suitable  steel  ratio  P  =  TSQ 
700/12,000^0.0109.  Adopting  0.011,  then  8'  gives  bd2=  135,000/^700  = 
1157.  Taking  6  =  7  in.  as  before,  d2  =  1157/7  =  165.3,  or  d=12.8,  13  in. 
say.  Finally  A  =  0.01  1  X  7  X  1  3  =  1  .00  in2. 

The  construction  of  the  diagram  -i  (Plates  I  -IV,  pages  213  to 
216)  referred  to  will  now  be  explained  and  then  their  use.  It  will 
be  convenient  to  have  names  for  the  quantities  }8pj  and 


FORMULAS  FOR  WORKING  LOADS.  61 

eqs.  3  and  4)  and  single  symbols  for  them.  We  shall  call 
them  coefficients  of  resistance  relative  to  the  steel  and  the  concrete 
and  will  denote  them  by  R8  and  Rc  respectively;  that  is, 

(a)     R.=f9pj    and     (b)    Rc  =  %fckj. 

Then  the  formulas  for  resisting  moments  of  a  given  beam 
with  particular  working  strengths  fs  and  Jc  may  be  written  thus  : 

M8  =  R8bd2    and    Mc=Rcbd2.     .    ...     (1) 

Similarly  for    any  particular  beam  subjected  to  a  bending 
moment  M, 

;.....    .     (2) 


Likewise   for  any  particular  bending   moment   and  working 
strengths  /8  and  fc,  the  necessary  section  is  given  by 

bd2  =  M/R}     .    .    ......     (3) 

R  being  the  smaller  of  the  two  coefficients  of  resistance. 

In  the  four  diagrams  values  of  p  are  given  at  the  upper 
and  lower  margins  and  values  of  R8  and  Rc  at  the  sides.  The 
diagrams  are  drawn  for  four  different  values  of  n,  i.e.,  10,  12, 
15,  and  18,  as  noted  on  the  plates. 

The  /,  curves  of  the  diagrams  are  merely  the  plots,  or 
graphs,  of  equation  (a)  for  certain  values  of  /«  as  marked  on 
the  curves.  The  fe  curves  are  the  graphs  of  equation  (b)  for 
various  values  of  fe  as  marked.  For  example,  when  n=15, 
/.  =  14,000,  /c=600,  and  p-1%  (see  page  215),  #,=  120  and 


The  foregoing  three  examples  will  now  be  solved  by  means 
of  the  diagram,  page  215  (ft  =  15). 

(1)  The  percentage  of  steel  being  1.26,  we  first  find  that  value  on 
the  lower  margin  ;  then  trace  vertically,  stopping  at  the  first  of  the  two 
curves  fc  =  600  and  /«=  15,000;  then  trace  horizontally  to  either  side 

*  These  diagrams  are  modeled  after  those  contributed  by  Prof.  French  in 
Trans.  Am.  Soc.  C.  E..  Vol.  LVI,  1906,  pp.  362-4. 


v 


62 


GENERAL  THEORY. 


[Cn.  III. 


margin  and  read  off  the  value  #  =  115.  Finally  M  =  115XlOxl42  = 
225,400  in-lbs. 

(2)  R  =  M/bd*  =  350,000/10  -172=  121,  and  the  percentage  of  steel  is 
1.04.    We  enter  the  diagram  with  these  values  of  R  and  p,  find  the  inter- 
section of  the  horizontal  and  vertical  lines  through  these  values  respect- 
ively, and  from  the  steel  and   concrete  curves  adjacent  to  this  inter- 
section estimate  /8  to  be  13,750  and  fc  675  lbs/in2. 

(3)  We  first  find  the  intersection  of  the  curves /f  =  700  and  fs=  12,000; 
from  that  point  tracing  down  we  find  p  =  1.35%,  and  tracing  horizontally 
we  find  R  =  137.    Then  bd2  =M/R  =  135,000/137  =  986,  from  which  b  and 
d  may  be  decided  upon,  and  then  finally  the  amount  of  steel. 

^  60.  Flexure  Formulas  for  Ultimate  Loads,  Based  on 
Parabolic  Variation  of  Compression  and  Neglecting  Tension 
in  Concrete. — It  is  assumed  that  the  amount  of  reinforce- 
ment is  sufficient  to  develop  the  full  compressive  strength 
of  the  concrete  without  straining  the  steel  beyond  its  yield 
point;  or  otherwise  expressed,  failure  occurs  by  crushing  of 
the  concrete,  the  stress  in  the  steel  being  still  within  the 
yield  point.  Then  the  parabola  representing  the  variation  of 
compression  is  a  full  parabola  (see  Art.  28),  the  upper  end 
(see  Fig.  19)  being  the  vertex. 


FIG.  19. 

If  the  amount  of  steel  in  a  beam  is  such  that  the  ultimate 
strength  of  the  concrete  and  the  elastic  limit  of  the  steel  would 
be  reached  simultaneously  if  the  beam  were  subjected  to  a  gradu- 
ally increasing  load,  then  this  will  be  called  the  ideal  amount — 
no  better  term  seems  available — but  this  amount  may  not  be 
the  best  in  a  given  case. 


§61.]  FORMULAS  FOR   ULTIMATE  LOADS.  63 

In  the  present  connection,  the  two  following  properties  of 
a  parabola  like  that  of  Fig.  19  are  useful:  (1)  The  average 
abscissa  of  the  parabolic  arc  equals  two-thirds  the  greatest,  /c; 
(2)  the  distance  from  the  centroid  of  the  parabolic  area  to 
its  top  equals  three-eighths  the  total  height,  kd. 

61.  Neutral  Axis  and  Arm  of  Resisting  Couple.  —  The  "initial 
modulus  of  elasticity"  of  the  concrete  (Art.  24)  is  denoted  by 
Ec  in  the  present  article.  It  is  represented  by  the  tangent 
of  the  angle  between  the  vertical  through  N  and  the  tangent 
to  the  stress-strain  curve  at  N.  And  since  NA  represents  ec, 
it  follows  from  a  well-known  property  of  the  parabola  that 
fc=%Ecec.  Also  fs  =  Ese8,  and  from  the  assumption  of  plane 
sections  it  follows  that  es/ec=(d—kd)/kd.  Eliminating  e8/ec 
from  the  above  equations,  and  introducing  the  'abbreviation  n, 
gives 


2n/c~    k 

When  the  loads  and  reactions  are  vertical  —  beam  horizontal  — 
the  total  tension  and  the  total  compression  on  the  section 
are  equal,  i.e., 


Eliminating  f8/fc  between  equations  (a)  and  (b)  .  and  intro- 
ducing the  abbreviation  p,  gives  3pn  =  k2/(l  —  k);  this  if  solved 
for  k  gives  _ 

?pn  ......     (1) 


This  formula  shows  that  the  neutral  axes  of  all  beams  of  a 
given  concrete  and  of  a  given  percentage  of  reinforcement 
are  at  the  same  proportionate  depth,  k,  for  their  respective 
ultimate  loads.  The  lower  group  of  curves  (Fig.  20)  gives  k 
for  different  values  of  p  and  n;  thus  for  p  =  2%  and  ^  =  15, 
&-0.60.  The  curves  show  that  k  increases  as  p  or  n  increases. 
The  distance  of  the  centroid  of  the  compressive  stress  from 
the  compressive  face  of  the  beam  is  f&d;  therefore  the  arm 
of  the  resisting  couple  TC  is  given  by 

jd  =  d-$kd,     or    /=!  -ffc:      ....     (2) 


64 


GENERAL  THEORY. 


[Cn.  in. 


i.oo 


05$ 


164 


.IX)- 


-30 


\ 


\ 


-40 


-50 


cu 


.00 


tM 
< 
Pi 


ss 


05'$, 


Percentage  of  S 


3  of  Steel 


16* 


FIG.  20. 


§  63.]  FORMULAS  FOR  ULTIMATE  LOADS.  65 

Plainly,  as  k  increases  j  decreases,  but  not  at  the  same  rate. 
The  upper  group  of  curves  in  Fig.  20  gives  j  for  different  values- 
of  p  and  n;  thus  for  p  =  2%  and  ft  =  15,  /  =  0.775.  It  should 
be  noticed  that  /  does  not  vary  much  with  p,  and  that  for 
n  =  15  and'  p  greater  than  1%  the  average  value  of  j  is  about 
0.80. 

62.  Ultimate  Resisting  Moment  for  a  Given  Ultimate 
Strength  fc.  —  Remembering  the  assumption  made  at  the  outset 
in  regard  to  the  amount  of  steel  (Art.  60),  it  will  be  under- 
stood that  the  ultimate  resisting  moment  always  depends  on 
the  concrete;  the  value  is 


d  =  $jkfJ>d*.'.'.    .    .     (3) 

It  should  be  remembered  that  this  equation  gives  the  ultimate 
resisting  moment  only  if  when  the  unit  stress  in  the  concrete 
is  at  the  ultimate  that  in  the  steel  is  not  beyond  the  elastic 
limit. 

.  If  the  beam  has  the  "ideal  amount"  of  reinforcement 
before  referred  to,  then  the  ultimate  resisting  moment  can  be 
computed  from  the  steel  by  means  of 

M  =T.id=f^-d=f.p]bd*,      ....     (4) 


in  which  /,  denotes  elastic  limit  of  steel. 

For  approximate  computations  one  may  use  the  average- 
values  /=0.80  and  &=0.52;  with  these,  formulas  (3)  and  (4) 
become  respectively 

Mc  =  Q.278fcbd2,      ......     .     (3)' 

(4)' 


63.  Determination  of  Amount  of  Steel-  and  Cross-section  of 
Beam  for  a  Given  Ultimate  Bending  Moment.  —  When  a  beam 
contains  the  "ideal  amount"  of  steel,  the  values  of  M  given 
by  (3)  and  (4)  are  equal;  hence,  fs/fc  =  2k/3p.  If  the  value 


66 


GENERAL  THEORY. 


[Cn.  III. 


of  k  as    given  by  equation  (a)  be  inserted  in  this  equation, 
then  the  following  formula  for  the  " ideal  steel  ratio"  results: 


2/3 


L>(-L 


(5) 


This  shows  that  p  depends  only  on  the  ultimate  strength  of 
concrete  and  elastic  limit  of  steel,  and  not  at  all  on  the  size 
of  beam.  Fig.  21  gives  graphically  the  " ideal  ratio"  p  for 


FIG.  21. 

different  values  of  the  ratio  f8/fc  and  four  values  of  n;    thus 
for  /.  =  34,000,  /c  =  1700,  and  n  =  15,  p  =  1.93%. 

If,  in  any  given  case,  the  steel  ratio  as  given  by  (5),  or  a 
higher  value,  is  adopted,  then  the  concrete  would  crush  without 
straining  the  steel  beyond  the  elastic  limit,  and  the  ultimate 
resisting  moment  of  the  beam  is  given  by  (3),  which  value 
equated  to  the  ultimate  bending  moment,  M,  to  be  provided 
for,  gives  %fcjkbd2  =  M,  or 


§64.]  FORMULAS  FOR  ULTIMATE  LOADS  67 

From  this  d  may  be  computed  for  any  assumed  value  of  b. 
If  a  lower  value  than  that  given  by  equation  (5)  is  adopted 
for  p,  then  under  a  gradually  increasing  load  the  stress  in 
the  steel  would  reach  the  elastic  limit  before  the  concrete 
would  crush,  and  the  formulas  of  this  article  could  not  be 
used  to  compute  the  ultimate  resisting  moment  of  the  beam. 
See  Art.  67  for  solution  of  this  case. 

Approximating  as  before,  /=0.80  and  A;  =  0.52,  and  eq.  (6) 
becomes 

.......     (6)'     \ 

64.  Diagrams  and  Examples. — Two  numerical  examples 
will  now  be  given  to  illustrate  the  foregoing  principles,  and 
then  a  diagram  will  be  explained  by  means  of  which  compu- 
tations in  such  examples  can  be  wholly  or  partially  avoided. 

(1)  A  concrete  beam  is  10X16  inches  in  cross-section  and  the  tension 
reinforcement  consists  of  four  f-in.  steel  rods,  their  centers  being  two 
inches  above  the  lower  face  of  the  beam.  The  ultimate  compressive 
strength  of  the  concrete  being  2000  and  the  elastic  limit  of  the  steel 
40,000  lbs/in2  compute  the  ultimate  resisting  moment  of  the  beam. 

Solutions.  Here  p  =  0.0126,  and  for  n=15,  eq.  (1)  gives  7c  =  0.52 
and  (2)  gives  /  =  0 . 805.  Hence 

Mc  =  l  0.805X0.52X2000X10X142  =  1,096,000  in-lbs. 

It  remains  to  test  whether  the  stress  in  the  steel  would  be  within  the* 
elastic  limit,  the  beam  being  subjected  to  a  bending  moment  of  1,096,000 
in-lbs.      This  is  done  by  dividing  the  bending  moment  by  the  arm  of  the 
resisting  couple,  which  gives  the  whole  tension  in  the  steel,  and  then 
this  tension  by  the  area  of  the  steel ;   thus 

1'°96'000    97,3001b,  =  r 


0.805  X 14 

and 

07  ^nn 

^  =  55,000  lbs./in'=/8 

This  result  being  beyond  the  stated  elastic  limit,  eq.  (3)  does  not  apply 
to  the  problem  in  hand.  (The  ultimate  resisting  moment  can  be  com- 
puted by  other  methods.  See  ex.  2,  page  76.) 


68  GENERAL   THEORY.  [Cn.  Ill, 

(2)  A  beam  is  to  be  figured  to  safely  withstand  a  bending  moment 
of  135,000  in-lbs.,  the  ultimate  compressive  strength  of  the  concrete 
being  taken  at  2000  and  the  elastic  limit  of  the  steel  at  40,000  lbs/in2. 

Solution.     With  n  =  l5,  eq.  (5)  gives  as  the  "  ideal  steel  ratio,"  since 


For  this  value  of  p,  eq.  (1)  gives  k  =  0.598,  and  (2)  gives  /=  0.775.  With 
a  factor  of  safety  of  3,  the  ultimate  bending  moment  is  405,000  in-lbs.,. 
and  eq.  (6)  gives 

,  „  __  405,000  __ 
~  f  X2000X  0.775X0.598" 

Trying  6  inches  for  b,  then  d2  =  109.3  or  d  =  10.5  in.  ;  also  A  =  0.02  X  6  X  10.5 
=  1.26  in2. 

The  "  coefficients  of  resistance"  on  the  parabolic  theory 
are  f8pj  and  %fcjk  (see  equations  4  and  3),  and  using  the  sym- 
bols Rs  and  Rc,  as  in  Art.  59, 

Rs  =  fapj    and     Rc  =  §/c/&. 

The  f8  curves  of  the  diagram  (Plate  V,  page  217)  are  graphs  of  the 
first  equation  for  certain  values  of  fs  as  marked  on  the  curves 
and  n  =  15.  (The  curves  for  n  =  12  differ  very  little  from  these  .) 
The  fc  curves  are  graphs  of  the  second  equation  for  various 
values  of  fc  as  marked;  the  full  curves  are  for  n  =  15  and  the 
dotted  for  n  =  12. 

In  using  the  diagram  to  determine  (1)  the  ultimate 
resisting  moment  of  a  given  beam  for  a  specified  ultimate 
compressive  strength  of  the  concrete,  or  (2)  a  steel  ratio  and 
size  of  beam  to  withstand  a  given  ultimate  bending  moment 
with  specified  compressive  strength  of  concrete,  these  formulas 
respectively  should  be  borne  in  mind: 

M  =  Rbtf    and.  bd2=M/R. 

The  foregoing  two  examples  will  now  be  solved  by  means  of 
the  diagram. 

(1)  The  percentage  of  steel  being  1.26,  we  first  find  that  value  on. 
the  lower  margin  of  the  diagram,  and  then  trace  vertically  to  the  line 
marked  /c  =  2000.  We  note  that  the  point  thus  found  is  above  the  line 


§65.] 


GENERAL  PARABOLIC  FORMULAS. 


69 


/8  =  40,000,  the  elastic  limit  of  the  steel  of  the  beam,  and  hence  conclude 
that  the  amount  of  steel  in  this  beam  is  insufficient  to  develop  the 
full  compressive  strength  of  the  concrete  without  straining  the  steel 
beyond  the  elastic  limit.  If  the  elastic  limit  of  the  steel  were  as  high 
as  55,000  lbs/in2,  we  would  trace  horizontally  from  the  point  as 
found  above  to  either  side  of  the  diagram  and  read  #  =  552.  Then 
M  =  fl&cP  =  552  X 10  X142  =  1, 087,000  in-lbs.,  which  is  the  ultimate  resist- 
ing moment  of  this  beam  with  the  high  elastic  limit  steel. 

(2)  We  first  find  the  intersection  of  the  curves  /c  =  2000  and 
/s  =  40,000 ;  from  that  point  tracing  down  we  find  p  =  2%,  and  horizontally 
we  find  #  =  620.  Then  bd*  =  M/R  =  405,000/620  =  654,  from  which  6 
and  d  can  be  decided  upon,  and  finally  the  amount  of  steel. 

65.  Flexure  Formulas  for  any  Load  up  to  Ultimate, 
Based  on  Parabolic  Variation  of  Compression  and  Neglect- 
ing Tension  in  Concrete  (After  Talbot). — It  is  assumed  that 
the  stress  in  the  steel  is  not  above  the  yield  point.  The  par- 
abola representing  the  variation  of  compressive  stress  is  not  a 
^full  one",  that  is,  its  top  is  not  the  vertex,  see  Fig.  22,  unless 


kd 


Jd 


the  maximum  concrete  stress  is  at  the  ultimate  value.  As 
heretofore  fc  and  ec  will  denote  the  unit  stress  and  strain  re- 
spectively at  the  compressive  face  of  the  concrete,  and  as  in 
Art.  61,  Ec  will  denote  the  initial  modulus  of  elasticity  of  the 
concrete.  In  this  article  //  and  ecf  will  denote  these  same 
quantities  at  the  ultimate  stage  of  the  concrete,  and  q  will 
be  used  as  an  abbreviation  for  ec/ec'.  It  can  be  shown  from 


70 


GENERAL  THEORY. 


[Cn.  III. 


the  properties  of  a  parabola  that:    (1)  The  average  abscissa 
to  the  parabola  NB  is  (3  -q)/3(2  -q)  times  the  greatest  abscissa 


0.9 
0.8 
0.7 

|0.5 
£0.4 
0.3 
0.2 
0.1 

A 

/ 

^ 

^ 

/ 

X 

X 

/ 

X 

X 

/ 

X" 

X 

/ 

/ 

X 

s* 

/ 

/ 

X 

X* 

/ 

/ 

/ 

/ 

x 

X 

^ 

ts 

x 

X 

Values  of  /c-r/c' 

FIG.  23a. 


FIG.  23b. 

/c;    (2)  the  distance  from  the  centroid  of  the  parabolic  area 
to   the  top  AB  is  (4  — g)/4(3— 5)  times  its  height,  kd;   and  (3) 

(a) 


§  66.]  FLEXURE  FORMULAS  AFTER  TALBOT.  71 

Fig.  23a  shows  graphically  the  relation  between  q  and  the 
ratio  tc/fc';  thus  when  q  =  \  (the  concrete  is  strained  to  one- 
fourth  its  limit  of  compression)  the  unit  stress  in  the  concrete 
is  about  0.45  of  the  ultimate  strength. 

The  lines  NB  in  Fig.  236  show  the  distributions  of  com- 
pressive  stress  at  a  section  of  a  beam  when  q  is  J,  J,  f  and  1 
respectively  as  marked.  In  each  case  N  is  the  neutral  axis 
and  AB  represents  the  unit  stress  on  the  remotest  fiber.  When 
q  is  i,  the  distribution  is  almost  linear. 

66.  Neutral  Axis  and  Arm  of  Resisting  Couple.  —  As  in 
Arts.  55  and  61,  es/ec  =  (d  —  kd)/kd,  and  fs  =  Eses.  Eliminating 
es/ec  from  these  two  equations  and  (a),  and  introducing  the 
abbreviation  n}  gives 


nfc    k(2-q 


When  the  loads  and  reactions  are  vertical  —  beam  horizontal  — 
the  total  tension  and  total  Compression  on  the  section  are 
equal,  i.e., 

Af.=bkdfe(3-q)/3(2-q).    .     .     .     .     .     (c) 

Eliminating  the  ratio  fs/fc  between  equations  (6)  and  (c),  and 
introducing  the  abbreviation  p,  gives  6pn(l  —  k)  =  k2(3  —  q), 
which  solved  for  k  furnishes  the  following  formula: 


m 


It  shows  that  the  neutral  axes  of  all  beams  of  a  given  con- 
crete and  a  given  percentage  of  reinforcement  are  at  the  same 
proportionate  depth,  k,  for  any  particular  stage  of  loading 
as  given  by  q.  The  lower  group  of  curves  in  Fig.  24  shows 
how  k  depends  on  p  and  n  for  q  =  \,  the  value  taken  by  Talbot 
as  closely  corresponding  to  the  working  stage.  The  lower 


72 


GENERAL  THEORY. 


[CH.  in. 


FIG.  24. 


§  68.]  FLEXURE  FORMULAS  AFTER  TALBOT.  73 

group  of  curves  in  Fig.  25  shows  how  k  depends  on  q  (that  is, 
on  the  stage  of  loading)  for  several  values  of  p,  n  being  taken 
as  15.  Thus  when  p  =  0.01  and  g  =  0  nearly  (load  very  small), 
^=0.42  and  when  5  =  1  nearly  (ultimate  load),  k  =  0.48. 

The  distance  of  the  centroid  of  the  compressive  stress  from 
the  top  of  the  beam  is  fcd(4  —  g)/4(3  —  q)\  hence  the  arm  of 
the  resisting  couple  is  given  by  jd  =  d  —  kd(4-q)/4:(3-q)  or 


The  upper  group  of  curves  in  Fig.  24  shows  how  j  depends 
on  p  and  n,  fqr  the  stage  q  =  \.  The  upper  group  of  curves 
in  Fig.  25  shows  how  j  depends  on  q  for  several  values  of  p, 
n  being  taken  as  15.  It  should  be  noticed  that  j  does  not 
change  much  for  considerable  changes  in  q. 

67.  Resisting  Moment  for  Given  Values  of  fc  and  fs. — Whether 
the  resisting  moment  is  determined  by  the  concrete  or  steel 
depends  on  the  percentage  of  reinforcement;  in  a  general 
way  the  higher  percentages  make  the  moment  depend  on 
the  concrete  and  the  lower  on  the  steel.  As  depending  on 
the  concrete,  the  resisting  moment  is  given  by 


The  value  of  q  to  be  used  here  must  correspond  with  the  fc 
used,  the  relation  between  q  and  fc  being  given  by  (a)  of  Art.  65. 
As  depending  on  the  steel,  the  resisting  moment  is 

(4) 


68.  Determination  of  Fibre  Stresses  fs  and  fc  for  a  Given 
Bending  Moment.  —  Formulas  for  these  can  be  obtained  by 
solving  (3)  and  (4)  for  fc  and  /.  respectively;  thus 

3(2-  g)    M 


I- #)  jkbd2 

t_ 

pjbd2 


(5) 


74 


GENERAL  THEORY. 


[Cn.  III. 


i.oo 


.90 


.70 


The  curves  give  values  of  J  and  k  according 
to  the  theory  of  Talbot;  the  heavy  horizontal  line: 
give  values  according  to  theory  based  on  the 
lineal  law  of  distribution  of  the  compressive 
fiber  stress. 


All  Values  of  j  and  k  a' 


bas 


id  on  w= 


Values  of    1 


PIG.  25. 


§69.]  FLEXURE  FORMULAS  AFTER  TALBOT.  75 

Neither  fc  nor  fa  can  be  determined  directly  from  these,  for  each 
formula  contains  q  (j  and  k  depend  on  q),  which  is  an  unknown 
in  the  problem  in  hand.  An  estimated  value  of  q  must  be  used 
for  a  trial  solution  of  (5),  and  then  with  the  value  of  fc  thus 
found  a  better  value  of  q  may  be  obtained  from  (a)  or  from 
Fig.  23,  which  value  may  be  used  in  a  second  trial  solution. 

69.  Determination  of  Amount  of  Steel  and  Cross-section  of 
Beam  fora  Given  Bending  Moment.  —  In  order  that  the  maximum 
unit  compression  in  the  concrete,  /c,  and  the  unit  stress  in  the 
steel,  fs,  may  have  certain  definite  values  when  the  beam  is 
subjected  to  a  given  bending  moment,  a  certain  definite  per- 
centage of  steel  must  be  used.  This  percentage  is  such  as 
makes  the  values  of  the  resisting  moment  as  determined  by 
steel  and  concrete  equal.  Thus  equating  values  of  M  from 
equations  (3)  and  (4)  and  simplifying, 

p=(3-q)kfc/3(2-q)fs. 

Inserting  in  this  the  value  of  k  furnished  by  (b)  gives 
3-g 


_ 
3(2-q)U2-qf. 

j\2n   /C 


In  this  also  the  value  of  q  used  should  correspond  to  the  value 
of  fc  adopted  as  working  stress.  The  curves  of  Fig.  26  give 
values  of  p  for  different  values  of  fs/fc  up  to  50,  q  being  taken 
atj. 

If  in  any  given  case  a  value  for  p  less  than  that  given  by 
(6)  is  adopted,  then  the  resisting  moment  is  given  by  equa- 
tion (4),  which  equated  to  the  bending  moment  to  be  provided 
for  gives  f8pjbd2  =  M,  or 

M>-£  .........     (7) 

fsP] 

If  a  greater  value  of  p  is  adopted,  then  the  resisting  moment 


76 


GENERAL  THEORY. 


[Cn.  III. 


is  given  by  (3),  which  if  equated  to  the  bending  moment  gives 
-q)  =  M,  or 


3-9 


(8) 


From  the   proper  one  of  these,  d  can  be  computed  for  any 
assumed  value  of  b. 


FIG.  26. 

Examples.  —  (1)  It  is  required  to  solve  example  1,  Art.  59,  by  the 
methods  of  this  article,  it  being  supposed  that  for  the  working  stress 
/c  =  600  lbs/in2,  q  =  ±. 

Solution.  As  shown  in  the  solution  of  the  example  referred  to, 
A  =  1.768  in2,  and  p  =  0.0126;  therefore  from  eq.  (1)  or  Fig.  24,  n  being 
taken  as  15,  A  =  0.466,  and  from  eq.  (2)  or  Fig.  24,  /  =  0.842.  Then  from 
eq.  (3) 


and 


Ms  -15,000  XI.  768x0.842x14  -313,000  in-lbs. 


(2)  It  is  required  to  solve  example  1  of  Art.  64  by  the  methods  of 
this  article. 

Solution.  As  disclosed  by  the  solution  in  Art.  64,  the  stress  in 
the  steel  will  reach  the  elastic  limit  before  that  in  the  concrete  would 


§70.]       FLEXURE  FORMULAS  AFTER  TALBOT.        77 

reach  the  ultimate  strength;  hence  the  ultimate  resisting  moment  de- 
pends on  the  steel.  The  stress  existing  in  the  concrete  when  the  steel 
is  stressed  to  the  elastic  limit  is  unknown  ;  so  is  q.  Supposing  that  this 
stress  in  concrete  is  f  the  ultimate  strength,  ^=0.5  (see  Fig.  23);  then, 
since  p  =  0.0 126,  and  n  is  taken  as  15,  A;  =  0.48  and  /  =  0.83  (see  Fig.  25), 
and  eq.  (4)  gives  Ms  =  820,000  in-lbs.  For  a  bending  moment  of  this 
value,  the  stress  in  the  concrete  would  be  (with  the  above  values  of  q,  /, 
and  k)  1260  lbs/in2  (see  eq.  5).  Now  for  the  ratio  1260/2000,  q  is 
about  0.4,  k,  0.75,  and  /,  0.825.  Since  this  value  of  /  is  practically  like 
the  one  used  in  the  trial  computation,  the  ultimate  resisting  moment  may 
be  taken  as  820,000  in-lbs. 

(3)  It  is  required  to  solve  example  2  of  Art.  59  by  the  methods  of 
this  article,  supposing  the  ultimate  compressive  strength  of  concrete  to 
be  2500  lbs/in2. 

Solution.  This  problem  can  only  be  solved  by  trial  because  it  is 
necessary  to  know  q  at  the  outset,  and  q  depends  on  a  quantity  sought, 
/c.  Supposing  that  the  load  is  about  a  safe  one,  then  q  equals  about 
£.  With  this  value,  n  equal  to  15,  and  p  equal  to  0.0104  (already  found 
on  page  60),  7c  =  0.43,  and  ;  =  0.85  (see  Fig.  24).  Then  eq.  (5)  gives 
/c  =  630  lbs/in2.  Now  q  depends  on  the  ratio  of  the  working  stress  in 
the  concrete  to  its  ultimate  strength;  for  the  approximate  value,  630, 
the  ratio  is  0.25,  and  eq.  (a),  or  Fig.  23,  gives  q  =  0.15.  With  this  value 
eq.  (1)  gives  k  =  0.432,  eq.  (2),  /  =  0.854  (see  also  Fig.  25),  and  eq.  (5), 
/c  =  635  lbs/in2.  This  value  is  so  near  the  first  that  #=0.15  must  be 
practically  correct,  and  /  =  0.854  may  be  used  to  determine  the  stress  in 
the  steel.  For  this,  eq.  (5)  gives  fa  =  13,700  lbs/in2. 

(4)  It  is  required  to  solve  example  .3  of  Art.  59  by  the  methods  of 
this  article,  the  ultimate  compressive  strength  of  the  concrete  being 
taken  at  2000  lbs/in2. 

Solution.  For  the  ratio  700/2000,  q  is  about  0.2  (see  Fig.  23).  With 
n  =  15  eq.  (6)  gives  p  =  0.018.  For  this  value  of  p,  we  may  use  either 
(7)  or  (8)  to  compute  the  dimensions  of  the  section.  Choosing  (7)  we 
need  first  a  value  of  /,  which  may  be  obtained  from  (2)  and  (1),  or  closely 
enough  from  Figs.  24  or  25;  the  figures  give  ;  =  O.S2,  and  eq.  (7)  gives 
fed2  =  763.  With  b  =  7  (as  in  Art.  59)  d  is  10.5  in. 

70.  Comparison  of  Flexure  Formulas  after  Talbot  with  (1) 
those  for  working  conditions  as  given  in  Art.  54-59,  and  (2) 
those  for  ultimate  conditions  as  given  in  Art.  60-64: 

(1)  The  heavy  horizontal  lines  of  Fig.  25  give  values  of  j  and 
k,  according  to  the  lineal  law  (Art.  54),  and  the  curved  lines 


78 


GENERAL  THEORY. 


[CH.  III. 


those  after  Talbot.  For  q  =  0.25  and  p  =  0.015,  the  difference 
between  the  two  values  of  /  is  represented  by  aa,  and  the  dif- 
ference between  the  two  values  of  k  by  b.  For  all  values  of  q 
up  to  0.25  or  0.30  the  first  difference  is  small,  and  so  the  values 
given  by  the  two  formulas  for  fs  must  be  nearly  the  same.  The 
second  difference  is  larger,  and  the  two  formulas  for  /«.  will  not 
agree  so  closely.  An  exact  comparison  will  now  be  made. 
Art.  56  gives  (see  eqs.  3  and  4). 


M 


pj'bd2 


and 


2M 

j'k'bd2' 


(The  primes  are  used  to  distinguish  the  symbols  from  the  cor- 
responding ones  in  the  other  formulas.)  Comparing  these  with 
eqs.  (3)  and  (4),  Art.  62,  one  gets 

//_  //_2(3-<?)  jk 

fs      f  fc      3(2-q)fk" 

As  already  explained,  q  rarely  exceeds  J  for  working  conditions ; 
with  this  value  and  n=15,  the  following  table  gives  the  ratios 
fs ' /fs  and  fc/fc  f°r  fiye  percentages  of  steel.  For  values  of  q 
less  than  J,  the  ratios  are  nearer  unity;  for  <?  =  0,  they  are  all 
unity  and  the  two  sets  of  formulas  are  identical. 


p  = 

}%. 

*%. 

1%. 

1.5%. 

2%. 

fs7fs 
fc7fc 

0.995 
1.092 

0.993 
1.091 

0.991 
1.090 

0.990 
1.088 

0.989 
1.086 

The  unit  stresses  in  the  steel  as  given  by  the  two  formulas  are 
practically  identical.  Any  error  involved  in  the  formulas  for 
//,  based  on  the  linear  law,  is  on  the  side  of  safety. 

(2)  For  loads  which  stress  the  concrete  to  the  ultimate 
limit,  the  stress  parabola  of  Fig.  22  is  full  like  that  of  Fig.  19, 
and  q  =  l.  The  formulas  of  Arts.  65-70  for  this  stage  and  those 
of  Arts.  60-64  are  identical. 

71.  Flexure  Formulas  for  T-beams.— The  following  dis- 
cussion is  based  on  the  linear  law  of  compression,  and  it  neglects 


§  72.] 


FORMULAS   FOR  T-BEAMS. 


the  tension  in  the  concrete.    The  following  additional  notation 
is  employed  (see  also  Fig.  27) : 
b   denotes  width  of  flange; 


effective  depth  of  beam; 
'       width  of  web ; 
'       thickness  of  flange; 

depth  of  neutral  axis  below  top  of  flange; 
'          '•'      "  resultant    compression    below    top 
flange. 

It  is  necessary  to  distinguish  two  cases,  namely,  (1)  the 
neutral  axis  is  in  the  flange,  (2)  the  neutral  axis  is  in  the  web. 
Which  of  the  two  cases  is  at  hand  in  any  particular  computa- 


d 
bf 
i 
c 
x 


of 


< — b--> 


tion  is  generally  not  apparent  at  the  outset.  One  may  assume 
either  case  and  determine  the  neutral  axis  on  that  basis,  and 
finally  note  whether  or  not  the  neutral  axis  falls  as  assumed. 
If  not,  the  computation  is  to  be  repeated  by  the  formulas  of 
the  other  case. 

72.  Case  I.  The  Neutral  Axis  in  the  Flange. — All  formulas 
of  Arts.  54-58  (except  approximate  ones)  apply  to  this  case. 
It  should  be  remembered  that  b  of  the  formulas  denotes  flange 
— not  web — width,  and  p  (the  steel  ratio)  is  A  4-  bd,  not 
A  +  Vd  (see  Fig.  27). 

Approximate  Formulas. — Evidently  the  arm  of  the  resisting 
couple,  CT,  is  always  greater  than  d—  %t',  hence  the  following 
approximate  formulas  err  on  the  side  of  safety : 


and    }8  = 


80  GENERAL  "THEORY.  LCH.III. 

These  give  good  results.  There  are  no  satisfactory  correspond- 
ing formulas  based  on  concrete,  and  indeed  they  are  unnecessary, 
as  the  flange  of  a  T-beam  is  generally  more  than  strong  enough 
for  the  steel. 

73.  Case  II.  The  Neutral  Axis  is  in  the  Web.  —  The  amount 
of  compression  in  the  web  is  commonly  small  compared  with 
that  in  the  flange,  and,  for  simplicity,  it  will  be  neglected,  as  no 
great  error  will  result. 

Neutral  Axis  and  Arm  of  Resisting  Couple.  —  Just  as  in 
Art.  55, 


es~fs/Es    d-c   ' 
The    average    unit    compressive    stress    on    the    flange    is 

-(c—  JO  and  the  whole  compression  is  -(c—%t)bt.    And  since 
c  c  . 

the  whole  tension  and  whole  compression  on  the  section  are 
equal, 

.     ......     (6) 


Eliminating  ft/fc  between  equations   (a)  and   (b)  we  get  an 
equation  which  when  solved  for  c  gives 

2ndA  +  bt2 
~ 


n  having  been  substituted  for  ES/EC.    This  equation  shows 
that  as  d  or  A  increases,  c  also  increases. 

The  arm  of  the  resisting  couple  is  d—  x  (see  Fig.  27).  The 
distance  x  is  equal  to  the  distance  of  the  centroid  of  the  shaded 
trapezoid  from  the  top  of  the  beam,  that  is 

ijp 


3c-2tt_ 
X~  2c-t  3 


Resisting  Moment  and  Working  Stresses.  —  If  the  beam  is 
under-reinforced,  its  resisting  moment  depends  on  the  steel; 


§74.]  FORMULAS   FOR  T-BEAMS.  81 

if  over-reinforced,  on  the  concrete.     These  two  values  of  the 
moment  are  respectively 

Ms  =  Afs(d-x)  ) 


If  one  is  in  doubt  which  of  these  to  use  when  about  to  com- 
pute the  resisting  moment  of  a  given  beam  with  specified 
working  stresses,  then  both  values  should  be  computed  and 
the  smaller  taken  as  the  resisting  moment. 

The  unit  stresses,  /«  and  fc,  produced  by  a  certain  bending 
moment  M  in  a  given  beam  can  be  computed  from 


~d-x'      s~  A'        ~nd-c      *        .' 

Approximate  formulas  corresponding  to  (3)  and  (4)  can  be 
established  as  follows:  From  the  stress  diagram  in  Fig.  27,  it 
is  plain  that  the  arm  of  the  resisting  couple  is  never  as  small 
as  d—  j£,  and  that  the  average  unit  compressive  stress  is  never 
as  small  as  J/c,  except  when  the  neutral  axis  is  at  the  top 
of  the  web.  Using  these  limiting  values  as  approximations  for 
the  true  ones,  we  have  as  substitutes  for  (3)  and  (4) 


M  T  2C 


C-T  __  /  —  —      /  —  —  .    .          .     .    (4Y 

~  d-\V    Is    A'     h     bt 

The  errors  involved  in  these  approximations  are  on  the  side 
of  safety,  for  (3)'  gives  values  smaller  than  (3),  and  (4)'  larger 
ones  than  (4). 

74.  Either  Case  I  or  II.  To  Dimension  a  T-beam  for  Given 
Loads.  —  Generally  the  thickness  of  the  flange  has  been  pre- 
determined, and  the  requirements  are  depth  of  beam  d,  amount 
of  reinforcement  A,  and  breadth  of  web  b'.  Explicit  formulas 
for  these  are  unwieldy,  and  the  practical  procedure  is  to  assume 


82  GENERAL  THEORY.  [Cn.  III. 

a  depth  d,  then  calculate  the  amount  of  reinforcement  A, 
by  approximate  formulas  (6)  or  (7)  as  the  "case"  may  be, 
and  finally  determine  the  actual  compressive  stress  in  the 
concrete  for  comparison  with  the  working  strength. 

To  determine  whether  the  case  in  hand  is  (I)  or  (II),  one 
may  make  use  of  equation  (5),  obtained  from  equation  (a), 
which  gives  the  value  of  d,  for  which  the  neutral  axis  is  at 
the  junction  of  flange  and  web  for  the  adopted  working  stresses  : 


If  a  smaller  value  of  d  was  adopted  the  case  is  (I)  and  if  a 
larger  the  case  is  (II).  The  following  are  approximate  for- 
mulas for  the  area  of  the  steel  for  these  cases  respectively. 


(6) 
and  A  =  M/(d-Wf.  .......     (7) 

Each  formula  errs  on  the  side  of  safety,  but  (6)  gives  correct 
values  when  the  neutral  axis  is  at  the  union  of  flange  and  web. 
The  web  must  be  wide  enough  to  take  the  steel  and  to 
provide  certain  strength  as  explained  later  (page  184). 

Examples.  —  (1)  A  T-beam  has  the  following  dimensions:  6=48  in., 
£=6  in.,  d  =  22  in.,  and  &'  =  10  in.;  the  steel  consists  of  six  f-in.  rods. 
If  the  working  strengths  of  steel  and  concrete  are  15,000  and  600  lbs/in2 
respectively,  and  n=15,  what  is  the  safe  resisting  moment  of  the  beam? 

Solution.  The  area  of  the  steel  is  2.65  in2,  and  p  =  2.65/(48x22)  = 
0.0025.  Supposing  this  beam  to  fall  under  Case  I,  we  find  k  from  Fig.  17 
(or  eq.  (1),  Art.  55)  to  be  about  0.24;  hence  kd  =  5.3  in.,  and  the  neutral 
axis  is  in  the  flange,  that  is,  the  case  was  correctly  guessed.  Now 
;  hence  (see  eqs.  (3)  and  (4),  Art.  56) 


Ms  =  (15,000  X2.65)(0.92  X22)  =806,000  in-lbs., 
and  MC  =  300(5.3X48)(0.92X22)  =  1,545,000  in-lbs. 

The  safe  resisting  moment  hence  depends  on  the  steel,  as  it  usually 
does  in  T-beams.  The  approximate  formula  gives  Afs  =  795,600  in-lbs. 

(2)  Change  t  of  the  preceding  example  to  4  in.  and  find  the  safe 
resisting  moment. 


x-^ 

/  OF  THE 

('    UNIVERSITY  I 

V  ,  °F      J 

§74.]  FORMULAS  FOR  T-BEAMSO^  83 

Solution.     Evidently  this  beam  now  falls  under  Case  II.     Equation 
(1)  gives  c  =  5.44  in.  and  (2)  x  =  1.61  in.     From  (3) 

M8  =  15,OOOX2.65(22-1.61)  =  950,000  in-lbs.,  ^ 


and  Me  =  (600/5.44)3.44(48  X4)(22  -  1.61)  =  1,485,000. 

The  approximate  formulas  (3)'  give  M8  =  716,000  and  Mc  =  1,152,000 
in-lbs. 

(3)  Suppose  that  the  diameter  of  the  rods  in  example  (1)  is  %  in., 
and  that  the  beam  is  subjected  to  a  bending  moment  of  1,250,000  in-lbs. 
Compute  the  working  stresses  in  the  steel  and  concrete. 

Solution.  The  area  of  the  rods  is  3.61  in2,  and  p  =  3.61/(48x22)  = 
0.0034.  Supposing  the  case  to  be  I,  we  find  k  from  Fig.  17  to  be  about 
0.27;  hence  kd  =  5.94  in.,  and  the  neutral  axis  does  lie  in  the  flange. 
Nowy=l-P=0.91;  hence  (see  eqs.  (5)  and  (6),  Art.  57) 


2X17,200X0.0034 
and  /c  =  --  -  =  481  lbs/ma. 

(4)  Suppose  that  the  diameter  of  the  rods  in  example  (1)  is  1  in., 
and  that  the  beam  is  subjected  to  a  bending  moment  of  1,250,000  in-lbs. 
Compute  the  working  stresses  in  the  steel  and  concrete. 

Solution.  Equation  (1)  gives  c  =  6.  74  in.,  hence  the  beam  falls  under 
Case  II.  Equation  2  gives  a:  =  2.22  in.,  and  (see  eqs.  4) 


The  approximate  formulas  (4)'  give  /«  =  13,960  and  /c  =  457  lbs/in2. 

(5)  The  flange  of  a  T-beam  is  24  in.  wide  and  4  in.  thick.  The 
beam  is  to  sustain  a  bending  moment  of  480,000  in-lbs.,  the  working 
strengths  of  steel  and  concrete  being  respectively  15,000  and  500  lbs/in2. 
What  depth  of  beam  and  amount  of  steel  will  answer? 

Solution.  We  will  try  d  =  18  in.  Equation  (5)  gives  d  =  l2  in.,  and 
hence  for  the  trial  value  the  neutral  axis  falls  in  the  web  and  the  beam 
falls  under  Case  II,  Equation  (7)  gives  A  =  2  in2.  It  remains  to  be 
seen  whether  the  working  stress  in  the  concrete,  with  d  =  18  and  A  =2,  is 
within  the  specified  limit.  From  equation  (1)  we  get  c  =  4.13  in.,  and 
from  equation  (4),  /c  =  300  lbs/in2,  which  is  within  the  limit;  hence  so 


84  GENERAL  THEORY.  [Cn.  IIL 

far  as  fibre  stress  in  steel  and  concrete  are  concerned  these  values  of  d 
and  A  will  suffice. 

75.  T-beams  Double-reinforced.  —  T-beams  are  often  con- 
tinuous over  their  supports;  at  such  places  the  bending  moment 
is  negative,  and  the  flange  is  under  tension  and  the  lower  part 
of  the  web  under  compression.     Not  only  is  tensile  steel  pro- 
vided, but  some  steel  is  always  left  in  the  web  (see  Chap.  VII)  ; 
that  is,  the  beam  is  reinforced  in  compression,  and  is  said  to  be 
double-reinforced.      For  a  discussion  of   double-reinforcement 
see  the  following  articles  —  particularly  Art.  79  —  in  which  there 
is  explained  a  simple  method  for  determining  the  effect  of  the 
compressive  steel  on  the  stress  in  the  tensile  steel  and  the  com- 
pression in  the  concrete. 

76.  Beams  Reinforced  for  Compression.—  The  compression 
in  the  concrete  is  assumed  to  follow  the  linear  law  and  the 
tension  in  it  is  neglected;   the  formulas  then  apply  to  working 
conditions  only.    In  addition  to  the  notation  already  adopted 
(see  page  54),  let 

A'  denote   the  cross-sectional  area  of  the  compressive  re- 

inforcement ; 
p'  denote  the  steel  ratio  for  the  compressive  reinforcement, 

that  is  A'/bd; 

f8'  denote  the  unit  stress  in  the  compressive  reinforcement; 
Cf  denote  the  whole  stress  in  the  compressive  reinforcement  ; 
df  denote  the  distance  from  the  compressive  face  of  the 

beam  to  the  plane  of  the  compressive  reinforcement; 
x  denote  the  distance  from  the  compressive  face  to  the 

resultant  compression,  C+  C',  on  the  section  of  the  beam. 

77.  Neutral  Axis  and  Arm  of  Resisting  Couple,  —  From  the 
stress  diagram  (Fig.  28)  it  appears  that  }s/nfc=(d—kd)/kd,  or 


(D 


Similarly,  /.'/«/„  =  (kd-  d')/kd,  or 

/-«*-=£/,       ......     (2) 


§77.] 


BEAMS  DOUBLE-REINFORCED. 


85 


For  simple  flexure,  the  whole  tension  T  and  whole  compression 
C+C'  are  equal,  hence 

f.A  =  lfJbkd+f.'A'.    ...    .    .    .    .     (a) 

Inserting  the  values  of  }8  and  /,'  fro.m  (1)  and  (2)  in  (a)  gives 
an  equation  which  may  be  written  thus: 

'/d),  .     ....     (3) 


;~f* 


c  +  c' 


3  CJ. 


FIG.  28. 

and  from  this  the  neutral  axis  of  a  given  section  can  be  located. 
The  lower  group  of  curves  in  Fig.  29  gives  values  of  k  for  several 
values  of  p  and  all  values  of  p'  up  to  2%;  n  is  taken  at  15 
and  d'/d  as  1/10.  Thus  for  p  =  2%  and  p'  =  1.5%,  &  =  0.434. 
The  arm  of  the  resisting  couple  is  the  distance  between  T 
(see  Fig.  28)  and  the  resultant  of  the  compressions  C  and  C'. 
It  follows  from  the  principle  of  moments  and  the  law  of  dis- 
tribution of  stress  respectively  that 


+  C'/C 


' 


.     C' 
and         - 


from  which  x  can  be  computed  for  any  given  section, 
the   arm     d  =  d—x   or 


Finally 

(4) 


The  upper  group  of  curves  in  Fig.  29  gives  values  of  j  for  sev- 
eral values  of  p  and  all  values  of  pf  up  to  2%;  n  is  taken  at 
15  and  d'/d  at  1/10.  Thus  for  p  =  2%  and  p'  =  1.5%,  /=0.875. 


86 


GENERAL  THEORY. 


All  Values  of.  j  and  &  are  based  on 
M-15  and 


Percentage  of  Compressive  Steel 


FIG.  29. 


§78.]  BEAMS  DOUBLE-REINFORCED.  87 

78.  Resisting  Moment  and  Working  Stresses.  —  If  the  tensile 
reinforcement  is  low,  the  resisting  moment  depends  upon  it, 
and  is  given  by 

(5) 


If  the  compressive  reinforcement  is  low,  the  resisting  moment 
depends  upon  it  and  the  concrete,  and  is  given  by 

Mc  =  i/c&(l  -  %k)bd*  +  fs'p'bd(d-  df)  ; 

but  //  bears  a  certain  relation  to  fc  (see  eq.  2),  which  inserted 
in  the  preceding  equation  gives  finally 

Mc  =  [k(±-ik)  +  npf(k-d'/d)(l--d'/d)/k]fcbd2.  .     .     (6) 

The  unit  fibre  stress  in  the  tensile   steel  produced  by  any 
bending  moment  M  can  be  computed  from 


__ 
pjbd2' 


and  those  in  the  concrete  and  compressive  steel  from  J8  and 
equations  (1)  and  (2)  respectively. 

Fig.  29  shows  that  the  neutral  axis  is  nearer  the  compressive 
steel  (k  <  0.55)  unless  the  percentage  of  tensile  reinforcement 
is  quite  high  and  the  compressive  low;  thus  for  p  =  3%,  the 
neutral  axis  is  nearer  the  compressive  steel  unless  p'  is  less 
than  3/4%,  and  when  p  =  2%,  it  is  nearer  for  all  values  of 
pf.  Now  since  the  unit  stresses  in  the  tensile  and  compressive 
steels  are  as  the  distances  of  the  steels  from  the  neutral  axis, 
it  follows  that  the  unit  stress  in  the  compressive  steel  is  gen- 
erally less  than  that  in  the  tensile,  that  is  //</s. 

For  approximate  computations  one  might  use  the  average 
values  j  =  0.85  and  k  =  0.45  in  equations  (5),  (6),  and  (7); 
then  they  would  become  respectively 

(5)' 

.....     (6)' 
(7)' 


88  GENERAL  THEORY.  [Cn.  III. 

79.  Determination  of  Amount  of  Compressive  Reinforce- 
ment.— This  problem  presents  itself  as  follows:  From  the  cir- 
cumstances of  the  case,  the  beam  needs  so  much  tensile  steel 
that  the  compressive  concrete,  if  unreinforced,  would  be  stressed 
too  high,  and  it  is  necessary  to  employ  compressive  reinforce- 
ment to  reduce  the  stress  in  the  concrete;  the  percentage  of 
reinforcement  necessary  to  lower  the  stress  a  certain  amount 
is  desired. 

An  explicit  formula  for  this  percentage  is  too  cumbersome 
for  practical  use,  but  a  diagram  (Plate  VI,  page  218)  can  be 
constructed  from  which  -the  desired  quantity  can  be  easily 
determined.  The  construction  of  such  a  diagram  will  now  be 
explained. 

Let  fs  and  fc  denote  the  unit  stress  in  the  tensile  steel  and 
the  concrete  respectively,  kd  the  depth  of  the  neutral  axis, 
and  jd  the  arm  of  the  resisting  couple,  CT,  when  there  is  no 
compressive  reinforcement  (see  Fig.  16);  also  let  //,  /</,  k'd, 
and  fd  denote  the  same  quantities  when  there  is  compressive 
reinforcement.  Then 

f8kd  Mk 


-and  /„'  = 


n(d-kd)     jdAn(l-k)' 
fs'k'd  Mk' 


n(d-kfd)     j'dAn(\-k')' 


From  these  the  relative  reduction  in  fc  due  to  the  addition  of 
compressive  steel  is  found  to  be 

fc~fc'_  jk'l-k 

~~/T     ~Jk  \=k'- 

Since  j  and  k  depend  on  p,  and  j'  and  k'  on  p',  the  equation 
furnishes  the  relation  between  relative  reduction  in  concrete 
stress  and  the  percentages  of  steel.  The  relative  reduction 
(fc—fc)/fc  depends  largely  on  the  percentage  of  compressive 
jsteel  and  for  a  given  value  of  this  percentage  the  reduction  is 
practically  the  same  for  all  ordinary  percentages  of  tensile 
steel  (from  \  to  3%).  Plate  VI,  page  218,  gives  values  of 


§79.]  BEAMS  DOUBLE-REINFORCED.  89 

this  reduction  for  different  values  of  compressive  steel  from 
0  to  2%.  As  heretofore,  values  n  =  15  and  d'/d  =  l/10  were 
used. 

Addition  of  compressive  steel  reduces  the  stress  in  the  ten- 
sile steel.    The  relative  amount  of  this  reduction  is  given  by 


V        /.  f 

The  group  of  curves  (Plate  VI)  gives  this  reduction  in  per 
cent  (right-hand  margin)  for  different  percentages  of  tensile  and 
compressive  steels  as  noted.  (For  illustration  of  the  use  of 
this  diagram,  see  example  (3)  following.) 

Examples. — (1)  A  beam  of  which  6  =  12  in.,  d  =  lS  in.,  and  d' /d  =  l/10 
has  2£%  of  tensile  steel  and  1%  of  compressive.  If  the  working  strengths 
of  steel  and  concrete  are  15,000  and  600  lbs/in2  respectively,  what  is 
the  safe  resisting  moment  of  the  beam? 

Solution.    From  Fig.  29,  fc  =  0.5  and  ;=0.85;   therefore 

M,  =  15,000  X 0.025  X0.85  X 12  X 182  =  1,238,000  in-lbs., 
and 
MC=  (0.5X0.417  +  15X0.01  X 0.4  X  0.9/0.5)600  X 12  X182= 736,000  in-lbs., 

which  is  the  safe  resisting  moment. 

(2)  Suppose  that  the  beam  of  the  preceding  example  were  subjected 
to  a  bending  moment  of  1,000,000  in-lbs.    What  are  the  working  stresses 
/c,  /a,  and  /.'? 

Solution.  As  in  example  (1),  ft  =  0.5  and  /=0.85;  therefore  (see 
eq.  7)  /«  =  1,000,000/0.025X0.85X12X182  =  12,100  lbs/in2.  From  equa- 
tion (1),  /C  =  (12jl00x0.5)-^  (15X0.5)  =  810  lbs/in2,  and  from  equation 
(2),  /«'=  15  (0.4/0.5)810  =  9720  lbs/in2. 

(3)  In  a  certain  design  of  a  beam  it  is  necessary  to  use  2.5%  of 
tensile  steel  and  this  would  result   in  a  stress   of  1200  lbs/in2  in  the 
concrete;   it  is  necessary  to  reduce  this  to  900  by  adding  compressive 
steel.     How  much  additional  steel  is  required? 

Solution.  (See  Plate  VI.)  The  desired  reduction  of  the  compressive 
stress  is  25%.  We  find  this  value  at  the  left  side  of  the  diagram,  then 
trace  horizontally  to  the  concrete  curve,  and  then  down  to  the  lower 
margin,  reading  there  0.9%,  the  required  quantity.  From  the  last  point 
we  trace  up  to  the  2.5%  steel  curve  and  then  to  the  right  margin,  where 
we  note  about  4.5%  reduction  in  tensile  steel  stress  due  to  0.9%  com- 
pressive steel. 


90 


GENERAL  THEORY. 


[Cn.  IH. 


80.  Flexure  and  Direct  Stress. — When  the  resultant,  R,  of  • 
the  external  forces  acting  on  one  side  of  a  section  of  a  beam  is 
not  parallel  to  the  section,  then,  in  general,  there  exist  both 
direct  and  flexural  stresses  at  the  section.  The  exception 
obtains  when  the  resultant  passes  through  the  centroid  of  the 
section  (transformed,  as  explained  below,  if  the  section  is  rein- 
forced unsymmetrically) ;  in  this  exceptional  case  the  fibre 
stress  is  wholly  direct. 

In  concrete  work,  the  direct  stress  is  always  compressive. 
Combination  of  direct  compressive  and  flexural  stress  gives 
resultant  fibre  stress  which  is  either  (1)  all  compression  or  (2) 
part  compression  and  part  tension;  these  cases  are  discussed 
separately  below.  Whether  a  given  R  will  produce  fibre  stress 
falling  under  case  (1)  or  (2)  depends  on  the  eccentricity  *  of  R, 
the  relative  amounts  of  steel  and  concrete  at  the  section  and 
on  n.  If  the  reinforcement  is  symmetrical,  steel  imbedded  a 
depth  equal  to  1/10  the  whole  depth  of  beam,  and  n  is  15,  then 
for  eccentricities  lower  than  those  given  in  the  table,  case  (1) 
obtains,  and  for  higher  case  (2). 


p= 

0% 

i% 

1% 

u% 

2% 

e/h= 

t 

0.187 

0.202 

0.214 

0.224 

In  addition  to  notations  already  adopted,  the  following  will  be 
used  (see  Fig.  30) : 

R  denotes  the  resultant  of  all  the  external  forces  acting  on 
a  beam  on  either  side  of  the  section  under  considera- 
tion; 

e  denotes  the  eccentric  distance  of  R'}  that  is,  the  distance 
from  the  point  where  R  cuts  the  section  to  the  middle 
of  the  section ; 
N  denotes  the  component  of  R  normal  to  the  section; 


*  By  the  eccentricity  of  R  is  meant  the  ratio  of  the  distance  between 
the  centre  of  the  section  and  the  point  where  R  pierces  the  section  to  the 
whole  height  of  the  section. 


§  81.]  FLEXURE  AND  DIRECT  STRESS.  91 

M  denotes  bending  moment  at  the  section;   it  equals  Ne  or 
the  sum  of  the  moments  of  all  the  external  forces  about 
the  horizontal  line  through  the  middle  of  the  section, 
but  when  the  transformed  section  is  used,  the  moment 
axis  must  be  taken  through  its  centroid; 
A'  denotes  the  area  of  the  steel  nearer  the  face  of  the  con- 
crete most  highly  stressed; 
df  denotes  the  distance  from  that  face  to  the  plane  of  this 

steel; 

A  denotes  the  area  of  the  steel  at  the  other  face; 
d  denotes  the  distance  from  the  former  face  to  the  plane  of 

this  steel; 

h  denotes  the  whole  height  of  the  section; 
pf  denotes  the  steel  ratio  A'/bh; 
p  denotes  the  steel  ratio  A/bh', 
u  denotes  the  distance  from  the  face  most  highly  stressed 

to  the  centroid  of  the  transformed  section; 
At  denotes  the  area  of  the  transformed  section; 
It  denotes  the  moment  of  inertia  of  the  transformed  section 

with  respect  to  its  horizontal  centroidal  axis; 
Ic  denotes  the  moment  of  inertia  of  the  section  bh  with 

respect  to  that  axis;  and 
I8  the  moment  of  inertia  of  the  sections  of  the  steel  with 

respect  to  the  same  axis. 

81.  Transformed  Section. — By  the  transformed  section  of  a 
reinforced  concrete  beam  is  meant  the  actual  section  with  the 
areas  of  the  reinforcement  replaced  by  concrete  n-fold  and  in 
the  planes  of  the  reinforcement.  Thus  if  Fig.  30  a  represents 
an  actual  section,  306  represents  the  section  transformed,  the 
areas  of  the  upper  and  lower  wings  of  the  latter  section  being 
respectively  n  times  the  areas  of  the  upper  and  lower  rein- 
forcements. 

(A  prism  of  steel  of  a  given  area  and  one  of  concrete  of  n 
times  the  area  are  equally  rigid  as  regards  simple  tension  or 
compression;  hence  a  reinforced-concrete  beam  and  a  plain 
concrete  beam  whose  section  is  that  of  the  first  transformed  are 


92 


GENERAL  THEORY. 


[Cn.  III. 


equally  stiff  in  so  far  as  stiffness  depends  upon  fibre  stress,  and 
in  certain  cases,  as  stated  later,  the  fibre  stress  in  the  reinforced 
beam  can  be  computed  from  those  in  the  plain  concrete  beam. 
In  those  cases,  the  actual  section  and  the  transformed  section 
equivalent,  ideally  at  least.  Actually,  the  two  beams  are 


FIG.  30. 


not  equally  strong  because  of  dangerous  stresses  in  the  wings 
of  the  transformed  section.) 

Referring  to  Fig.  30  it  will  readily  be  seen  that 


It=Ic+nI., 


u  = 


h/2+npd+np'd' 
1  +np 


(1) 


(2) 


w)3]    and    I8  =  A(d—  u)2+A'(u—  d')2.     (3) 
If  the  reinforcement  is  symmetrical,  then  u=h/2  and 

and    I8  =  2A(%h-d')2.    . (3)' 


82.  Case  I.  The  Fibre  Stress  is  Wholly  Compressive. — (a) 
The  unit  fibre  stress  in  the  concrete  can  be  computed  just  as 
though  the  beam  were  homogeneous,  but  the  transformed 
section  must  be  used  in  the  computations  if  the  beam  is  rein- 
forced. The  unit  stresses  in  the  steel  will  be  n  times  those  in 
the  concrete  in  the  planes  of  the  reinforcements  respectively. 
Thus  the  unit  direct  stress  in  the  concrete  is  N/At'}  the  unit 
flexural  stress  in  the  concrete  highest  stressed  is  Mu/It]  that 
in  the  concrete  adjoining  the  reinforcement  highest  stressed 


|82.J 


FLEXURE   AND  DIRECT  STRESS. 


is  M(u—  d')/It',  and  that  in  the  concrete  adjoining  the  other 
reinforcement  is  M(d—u)/It.    The  combined  unit  stresses  are: 


N     Mu 

{c  -+-> 


N     nM(u-df) 
A+-  "7T| 

N     nM(d-u) 


(4) 
(5) 
(6) 


These  equations — and  the  stress  diagram,  Fig.  31 — show  that 
fa  is  always  less  than  //,  and  //  is  always  less  than  n/c;  hence 
the  unit  stresses  in  both  steel  reinforcements  will  always  be 
safe  if  fc  is  a  safe  value. 


FIQ.  31. 

(6)  The  method  employed  for  simple  flexure,  suitably  modi- 
fied, leads  to  formulas  not  involving  the  transformed  section, 
as  will  now  be  explained. 

From  the  stress  diagram  (Fig.  32)  it  will  be  seen  that 

f.'-nfe(l-d'/kh), (7) 

fa=nfc(l-d/kh), (8) 

and  c'-c(l-l/fc) (9) 


94  GENERAL  THEORY.  [Cn.  III. 

From  the  condition  that  the  resultant  fibre  stress  equals  N, 


and  from  the  condition  that  the   moment  of  the  total   fibre 
stress  about  the  centroidal  axis  equals  Af, 


From  these  equations  it  is  possible  to  compute  the  unit  fibre 
stresses  /c,  fs,  and  /,'  in  a  given  case. 

When  the  reinforcement  is  symmetrical  the  equations  simplify 
greatly,  and  they  lead  to  the  following  formula  : 


h2  +6(1  +2np)e/h;    .     (10) 
they  also  give  the  following  formula  for  fc  or  M  : 


When  d'A  =  l/10,  and  n  =  15,  Fig.  33  gives  values  of  1/k 
for  different  values  of  eccentricity  and  percentage  of  steel; 
thus  fore/A  =  0.1  and  p  =  1.5%,  l/k  =  0.635,  hence  A;  =  1.57. 

83.  Case  II.  There  is  Some  Tension  at  the  Section.  —  (a)  If 
the  tension  m  the  concrete  is  so  small  as  to  be  permissible,  and 
this  tension  is  taken  account  of  in  the  computations,  then  the 
unit  fibre  stresses  in  the  concrete  and  steel,  if  reinforcement 
is  present,  may  be  computed  by  the  method  explained  under 
Case  I.* 

The  combined  unit  stress  in  the  remote  tensile  fibre  is  given 

by 


h-u 


(12) 


*  It  is    assumed  that  the  linear  law  of  variations  of  the  unit  flexural 
stersses  holds  for  the  tension  as  well  as  compression. 


§83.] 


FLEXURE  AND  DIRECT  STRESS. 


95 


0.05 


0.10 


0.15 


x, 


zzz 


////L 


///// 


///t 


I 


ItilL 


/-// 


/// 


LL 


All  Values  of  fc  are 
based  on  n*=15  and 


/// 


i 


0.05 


0.10 
Eccentricity  e/& 

FIG.  33, 


0;15 


96 


GENERAL  THEORY. 


[Cn.  III. 


and  /,  as  given  by  (5)  is  compressive  or  tensile  according  as  its 
value  is  positive  or  negative. 

(&)  If  the  tensile  stresses  are  so  high  that  it  is  advisable  to 
neglect  the  tension  in  the  concrete,  then  a  method  similar  to 
that  used  heretofore  in  simple  flexure  is  simplest.  The  trans- 
formed section  is  not  used.  0  (Fig.  34)  denotes  a  horizontal 


FIG.  34. 

axis  at  mid-depth  of  the  beam,  M  the  moment  sum  of  all  the 
external  forces  on  one  side  of  the  section  with  respect  to  that 
axis,  and  N,  as  before,  the  algebraic  sum  of  the  components 
of  those  forces  perpendicular  to  the  section.  From  the  stress 
diagram,  it  follows  that 


and 


(13) 


Since  the  resultant  fibre  stress  equals  N, 


and  since  the  moment  of  the  fibre  stress  about  the  horizontal 
axis  through  0  equals  M, 


From  these  four  equations  k}  fc,  fa,  and  //  can  be  determined 
for  a  given  section,  reinforcement,  M,  and  N. 


§84.]  FLEXURE  AND  DIRECT  STRESS.  97 

//  the  reinforcement    is  symmetrical,   then    the    equations 
simplify.    The  value  of  k  is  given  by 


The  greatest  unit  compressive  fibre  stress  in  the  concrete  is 
iven  b 


given  by 


and  the  unit  stresses  in  the  steel  are  given  by  (7)  and  (8). 
From  (7),  or  the  stress  diagram,  it  is  plain  that  //  is  less  than 
nfc  even  for  unsymmetrical  reinforcements. 

When  d'/h  =  l/lQ  and  n  =  15,  Fig.  35  gives  values  of  k  for 
different  values  of  eccentricity  and  percentage  of  steel;  thus 
for  e/h  =  l,  and  p=0.8%,  &=0.42. 

84.  Diagrams.  —  To  facilitate  the  application  of  equation  (11) 
(Case  I)  and  equation  (16)  (Case  II),  Plates  VII  and  VIII, 
pages  219  and  220,  have  been  constructed. 

In  the  first  diagram,  values  of  the  eccentricity,  e/h,  are 
given  at  the  upper  and  lower  margins;  the  ordinates  from 
the  lower  margin  to  any  curve  are  values  of  (l  +  24.npa2/h2)/12k 
(see  equation  11),  and  hence  of  M/bh2fc  ,for  the  value  p  marked 
on  that  curve.  Thus  when  e/h  =  0.1  and  p  =  1%,  M/bh2fc  =0.087. 

The  dotted  portions  of  the  curves  correspond  to  eccen- 
tricities which  involve  small  tensile  stress  in  the  concrete  and 
belong  strictly  to  Case  II.  The  values  of  the  unit  tensile  stress 
}c  can  be  calculated  from  equation  (12)  or  from 


fc~    kh    ~k      ) 

l/k  being  obtained  from  equation  (10),  or  from  an  extension  of 
the  appropriate  curve  in  Fig.  33. 

In  the  second  diagram,  also,  values  of  the  eccentricity  e/h 
are  given  at  the  upper  and  lower  margins;  the  ordinsaet 
from  the  lower  margin  to  any  solid  curve  are  values  of 


98 


GENERAL  THEORY. 


[Cn.  III. 


Eccentricity,  e/1.,  for  Lower  Group  of  Curves. 
0.5  1 0  1.5 


2. 


All  Values  of  fc  are  based 

on  n=  15  and  d'-?h  =  Vic 


0.2  0.3 

Eccentricity,  e/h,for  Lower  Group  of  Curves. 

FIG.  35. 


§  85.]  FLEXURE  AND  DIRECT  STRESS.  99 

j%k(3-2k)+2pna2/kh2  (see  equation  16),  and  hence  of 
M/bh2fc,  for  the  value  of  p  marked  on  that  curve.  Thus 
when  e/h  =  l  and  p  =  l%,  M/W/C  =  0.187. 

The  dotted  curves  in  the  second  diagram  enable  one  to 
estimate  the  ratio  of  the  unit  stress  in  the  tensile  steel  to  that 
in  the  concrete,  /s//c;  for  most  eccentricities  and  percentages 
of  steel.  Thus  when  e/h  =  l  and  p=0.5%,  we  find  e/h  =  l 
at  the  top  or  bottom  and  then  trace  vertically  to  the  0.5% 
curve  and  note  the  point  of  intersection.  This  point  falls 
between  the  curves  /s//c=20  and  25,  and  the  ratio  is  about  21. 
For  values  of  e/h  and  p,  which  bring  the  "point"  to  the  left 
of  the  line  /«//c  =  15,  /,  will  be  less  than  15/c,  and  hence  less 
than  the  working  strength  of  steel  for  all  ordinary  allowable 
values  of  fc.  No  similar  curves  for  fs/fc  appear  on  the  first 
diagram  because  that  ratio  is  always  less  than  15,  and  hence 
the  unit  stresses  in  the  steel  (both  upper  and  lower)  are  within 
safe  values  for  Case  I,  if  fc  is  safe. 

85.  Examples.  —  It  is  supposed  in  these  that  the  steel  is 
imbedded  a  depth  of  one-tenth  the  total  height  of  the  beam, 
and  that  n  =  15,  so  that  the  diagrams  on  pages  219  and  220  apply. 

(1)  A  beam  is  12  in.  wide,  30  in.  high,  and  contains  1%  of  steel 
above  and  an  equal  percentage  below.  At  a  particular  section,  the 
resultant  R  is  80,000  Ibs.,  its  inclination  to  the  axis  of  the  beam  is  5°, 
and  its  eccentric  distance  is  4.5  in.  Compute  the  unit  fibre  stresses 
in  the  concrete  and  steel  (fc,  f8,  and  /«'). 

Solution.     The  eccentricity  is  e/h  =  0.15,  and  M  =  80,000  cos  5°  X  4.5  = 

358,650  in-lbs.     The  beam  falls  under  Case  I  because  this  eccentricity 

gives  a  "  point  "  on  the  1%  curve  of  page  219,  but  not  on  that  of  page  220. 

Tracing  horizontally  from  the  point  we  read  M/W/C  =  0.112;    hence 

358650 


The  unit  stresses  in  the  steel  are  less  than  15/c=4500  lbs/in2.  Their 
exact  values  can  be  computed  from  equations  (7)  and  (8);  the  value  of 
k  for  use  in  them  can  be  easiest  obtained  from  the  diagram  on  page  95. 

(2)  Change  the  eccentricity  of  the  preceding  example  to  15  in.  and 
solve. 

Solution.  The  eccentricity  is  e/h  =  0.5,  and  M  =  80,000  cos  5°  X  15  = 
1,195,500  in-lbs.  The  beam  falls  under  Case  II  (see  page  220),  and  for 


100  GENERAL  THEORY.  [Cn:  III. 

the  eccentricity  0.5  and  1%  of  steel  the  diagram  gives  M/bh%  =  0.171; 

hence 

1,195,500 

*-12X30'X0.171-6471b6/m- 

The  intersection  of  the  1%  curve  and  the  0.5  eccentricity  line  lies  to 
the  left  of  the  curve  fs/fc  =  15 ;  hence  the  unit  stress  in  the  tensile  steel 
is  less  than  15X647  =  9470  lbs/in2.  The  exact  value  can  be  computed 
from  equation  13;  the  value  of  k  for  use  in  it  can  be  obtained  easiest 
from  the  diagram  on  page  98. 

(3)  The  breadth  of  a  beam  is  12  in.  and  its  height  24  in.    At  a  certain 
section  the  bending  moment  is  450,000  in-lbs.,  and  the  eccentric  distance 
is  4  in.      The  working  strength  of  the  concrete  being  600  lbs/in*,  how 
much  steel  reinforcement,  if  any,  is  required? 

Solution.    The   eccentricity  is  e/h  =  %,  and  hence   the  beam  would 
be  on  the  border  between  Case  I  and  II  even  if  no  steel  were  used.    With 
steel,  the  beam  falls  under  Case  I,  and 
M_         450,000 
bh%     12  X  24' X  600 

Entering  the  diagram,  page  219,  with  this  value  and  tracing  horizontally 
to  the  0.167  eccentricity  vertical,  we  find  their  intersection  and  note 
that  it  falls  between  the  0.6  and  0.8%  curves;  about  0.7%  of  steel  there- 
fore is  required. 

(4)  In  example   (3)  change   the  eccentric  distance  to  12  in.  and 
solve. 

Solution.  The  eccentricity  is  e/h  =  %,  and  the  beam  falls  under 
Case  II  (see  page  220).  M/bh%  has  the  same  value  as  in  example  (3); 
hence  entering  the  diagram  with  that  value  and  tracing  horizontally 
to  the  0.5  eccentricity  vertical,  we  find  their  intersection  and  note  that 
it  falls  between  the  0.2  and  0.3%  curves;  hence  0.3%  is  the  required 
amount 

(At  first  thought  it  may  seem  that  more  steel  is  necessary  in  example 
(4)  than  in  (3)  because  of  the  greater  eccentricity  in  the  former  example. 
But  it  should  be  noted  that  the  thrust  N  is  much  less  in  (4)  than  in  (3), 
its  values  being  M /e  =  37,500  and  112,500  Ibs0  respectively.) 

86.  Shearing  Stresses  in  Reinfoioed  Beams. — In  Art.  46 
the  variation  in  shearing  stress  in  a  homogeneous  beam  was 
discussed  and  the  general  formula  given  for  the  intensity  of 
shear  at  any  point  (see  eq.  (1)).  In  a  reinforced  beam  the 
variation  in  shear  differs  from  that  in  a  homogeneous  beam 
owing  to  the  concentration  of  tensile  stress  in  the  steel.  The 


§86.] 


FLE'XURE  AND  DIRECT  STRESS. 


101 


general  formula  for  shearing  stress  may,  however,  still  be  used 
if  the  transformed  section  be  employed;  that  is,  if  the  area 
of  the  steel  be  multiplied  by  n  and  considered  equivalent  to 
concrete  at  the  same  horizontal  plane.  The  tension  area  of 
the  concrete  should  be  neglected.  A  simpler  method  for 
present  purposes  is  the  following:  In  Fig.  36  is  represented  a 
short  portion  of  a  beam  where  the  total  vertical  shear  is  V. 
Let  v = horizontal  (or  vertical)  shearing  stress  per  unit  area 
at  the  neutral  axis,  and  let  b= width  of  beam.  Other  quan- 
tities are  sufficiently  indicated  in  the  figure.  C=T  and  C'  =  Tr. 


Neutral  Plane 


_  Steel 


FIG.  37. 


The  total  shearing  stress  on  any  horizontal  plane  between  the 
steel  and  the  neutral  axis  will  be  equal  to  Tf—  T  and  the  stress 

T'  —  T 
per    unit    area  =  v  =  —rjr~  •    From   equality  of  moments  we 

have  the  relation  Vdl=  (Tf—  T)jd,  whence  is  derived  the  expres- 
sion 

V 


bjd' 


(1) 


The  shearing  stress  given  by  eq.  (1)  is  the  same  at  all  points 
between  the  neutral  axis  and  the  steel;  above  the  neutral  axis 
the  shear  follows  the  parabolic  law  as  in  a  homogeneous  beam. 
Fig.  37  represents  the  law  of  variation  for  the  case  under 
discussion. 

Using  7/8  for  an  approximate  value  of  /  (see  Art.  65)  we 
have  approximately 


7M' 


102  GENERAL  THEORY.  [Cjj.  IIL 

that  is,  the  shearing  stress  at  the  neutral  axis  (equal  to  the 
maximum)  is  one-seventh  or  about  14%  more  than  the  aver- 
age value. 

87.  Beams  Reinforced  for  Compression. — In  beams  reinforced 
for  compression  formula  (1)  will  still  apply,  the  value  of  jd 
being  the  distance  between  the  tensile  steel  and  the  resultant 
of  the  compressive  stresses  as  shown  in  Art.  77.      In  this  case 
j  is  somewhat  greater  than  in  the  previous  case  and  v  is  more 

V 
nearly  equal  to  the  average  shearing  stress  T-T. 

88.  T -beams. — Here  again  formula  (1)  still  holds  true,   / 
retaining  its  general  significance.     As  shown  in  Art.  73,  /  may 
be  taken  as  closely  equal  to  the  distance  from  the  steel  to 
the  centre  of  the  flanges  of  the  T;    hence 


(3) 


It  is  to  be  noted  that  in  the  T-beam  the  shearing  stresses 
are  practically  the  same  as  in  a  rectangular  beam  of  the  same 
depth  and  having  the  same  width  as  the  stem  of  the  T.  The 
slab  aids  in  reducing  the  shear  only  by  its  effect  in  increasing 
slightly  the  value  of  /. 

89.  Working  Formula.  —  Since  the  value  of  /  varies  only 
within  narrow  limits  it  is  quite  as  satisfactory  for  comparative 
purposes  and  for  purposes  of  design  to  use  the  average  value 
of  the  shearing  stress, 


hi  which  b  is  the  breadth  and  d  is  the  net  depth  of  the  beam. 
In  T-beams  b  is  the  breadth  of  the  stem  and  d  is  the  total 
depth  from  top  of  beam  to  steel.  The  true  maximum  shear 
will  generally  be  from  10  to  15  per  cent  higher  than  the  aver- 
age value  thus  determined. 

90.  Effect  of  Shear  on  the  Tensile  Stresses  in  the  Concrete.  —  In 
Art.  46  it  was  shown  that  in  a  homogeneous  beam  the  direc- 


§90.]  SHEARING  STRESS.  103 

tion  of  the  maximum  tensile  stresses  is  horizontal  at  the  lower 
face  and  becomes  more  and  more  inclined  as  the  neutral  axis 
is  approached,  reaching  an  inclination  of  45°  at  that  place. 
In  the  reinforced  beam  we  have  assumed,  for  purposes  of 
design,  that  there  is  no  tension  in  the  concrete.  While  such 
possible  tension  will  add  very  little  to  the  resisting  moment 
of  the  beam  it  is  desirable  to  consider  it  here  in  relation  to 
the  shearing  stresses  and  the  resultant  effect  on  lines  of  prob- 
able rupture.  The  shearing  stresses  determined  in  the  pre- 
ceding article  have  been  calculated  on  the  assumption  of  no 
tensile  stress  in  the  concrete,  but  the  effect  of  such  tension 
on  the  distribution  of  the  shear  is  very  small  and  need  not  be 
considered. 

To  determine  the  amount  and  direction  of  the  maximum 
inclined  tensile  stresses  at  any  point,  eq.  (1),  Art.  46,  is  still 
applicable.  In  this  case  large  shearing  stresses  exist  imme- 
diately above  the  steel,  hence  the  maximum  tensile  stresses 
become  considerably  inclined  as  soon 'as  we  leave  the  line  of 
the  steel,  the  exact  direction  depending  upon  the  relation 
between  the  shear  and  the  horizontal  tension.  Exact  calcula- 
tions are  impossible,  since  the  actual  horizontal  tension  in  the 
concrete  is  unknown.  While  the  steel  is  assumed  to  carry  all 
tension  the  concrete  will  in  fact  be  stressed  in  accordance 
with  its  deformation  up  to  the  point  of  ultimate  deformation 
and  rupture.  Where  the  steel  has  a  stress  of  its  full  working 
value  of  12,000  to  15,000  lbs/in2,  the  deformation  will  much 
exceed  the  ultimate  deformation  of  the  concrete  and  rupture 
must  occur,  but  at  points  where  the  steel  stress  is  low,  as  for 
example  near  the  end  of  the  beam,  the  concrete  may  be  intact. 

Suppose,  for  example,  the  stress  in  the  steel  is  3000 
lbs/in2.  If  the  modulus  of  elasticity  of  the  concrete  in  ten- 
sion is  1,500,000  the  stress  in  it  will  be  3000/20  =  150  lbs/in2, 
which  is  not  far  from  its  ultimate  strength.  Suppose  further 
that  the  unit  shearing  stress  in  the  lower  part  of  the  beam  is 
100  lbs/in2.  By  eq.  (1)  of  Art.  46  the  resultant  maximum 
tension  will  be  *=i(150)+vVl502+1002=200  lbs/in2,  and 


104  GENERAL  THEORY.  [Cn.  III. 

will  have  a  direction  inclined  26J°  from  the  horizontal.  This 
stress  may  exceed  the  ultimate  strength  of  the  concrete  and 
the  result  will  be  an  inclined  crack.  At  points  nearer  the 
neutral  axis  the  horizontal  tensile  stresses  become  less  and 
the  inclined  tension  approaches  the  value  of  the  shearing 
stress  and  its  inclination  approaches  45°.  The  result  of  these 
inclined  stresses  is  likely  to  be  a  progressive  tension  failure 
in  an  inclined  direction  which  the  horizontal  rods  are  not  very 
effective  in  preventing. 

Excessive  stresses  of  this  kind  are  prevented  in  various 
ways.  Obviously  they  will  be  reduced  by  keeping  the  hori- 
zontal tension  small  through  the  use  of  considerable  horizontal 
steel  at  points  of  heavy  shear,  by  keeping  the  shearing  stresses 
low,  and  by  various  means  of  directly  carrying  the  inclined 
stresses  by  special  reinforcement. 

91.  Ratio  of  Length  to  Depth  for  Equal  Strength  in  Moment 
and  Shear.  —  For  any  given  values  of  per  cent  of  steel  and  of 
working  stresses  in  shear  and  direct  stress  there  is  a  definite 
ratio  of  length  to  depth  of  beam  which  will  give  equal  strength 
in  moment  and  shear.  The  strength  of  beams  of  greater  rela- 
tive length  will  be  determined  by  their  moment  of  resistance, 
while  that  of  shorter  beams  by  their  shearing  resistance.  The 
ratio  of  length  to  depth  for  equal  strength  depends  on  the 
method  of  loading. 

For  Single  Concentrated  Loads.  —  In  this  case  the  shear  V,  due 
to  a  given  load  W,  is  JTF,  and  the  moment  M  is  \Wl.  Hence 

W  =  2V  =  4M/l  .......     (a) 

From  Art.  89  we  have  V  =  v'bd  and  from  Art.  56 
M8  =  fajpbd2,  in  which  v'  =  safe  average  shearing  stress  and 
f8—  working  stress  in  steel.  Substituting,  we  have 


from  which 


§93.]  SHEARING  STRESS.  105 

For  a  Uniformly  Distributed  Load  a  similar  process  gives  the 
ratio 


d     v'  •    :  •  /.AI:-  •  • 

For  Beams  Loaded  with  Equal  Loads  at  the  Third  Points, 

I     3/,/p  m 

%-^r-     v  -•   •,••;•••    (3) 

In  the  case  of  continuous  girders  these  formulas  will  apply 
if  I  be  taken  as  the  length  between  points  of  inflection. 

Taking,  for  example,  p-0.01,  ?/  =  50  lbs/in2,  and  /„  =  15,000 
lbs/in2,  and  using  an  average  value  of  7/8  for  /,  we  have  the 

following  ratios  for  -r: 

For  concentrated  loads  -r—  5.25. 

a 

For  uniformly  distributed  loads  -7  =  10.5. 

For  double  concentrated  loads  -7  =7.87. 

92.  Bond  Stress.  —  The  stress  on  the  bond  between  steel 
and  concrete  (Fig.  36,  Art.  86)  will  be  equal  to  T'-T  on 
the  length  dl. 

If  U  denote  the  bond  stress  per  lineal  inch,  we  then  have 


T'—T 

u=~~dT' 


whence  we  derive 


The  bond  stress  per  unit  area  will  be  equal  to  U  divided  by 
the  sum  of  the  perimeters  of  the  steel  sections. 

93.  Strength  of  Columns. — Concrete  columns  need  rarely 
be  calculated  as  long  columns.  In  ordinary  construction 
the  ratio  of  length  to  least  width  will  seldom  exceed  12  or  15, 
while  the  results  of  tests  indicate  little  or  no  difference  in 


106  GENERAL  THEORY.  [Cn.  III. 

strength  for  ratios  up  to  20  or  25.  It  will  be  desirable  then 
to  determine  first  the  strength  of  a  reinforced  column  con- 
sidered as  a  short  column.  If  the  conditions  require  it  a  gen- 
eral column  formula  may  then  be  applied  to  provide  for  cases 
where  the  length  is  excessive. 

94.  Methods  of  Reinforcement. — Columns  are  reinforced  in 
two  ways:    (1)  by  means  of  longitudinal  rods  extending  the 
full  length  of  the  column,  and  (2)  by  means  of  bands  or  spirally 
wound  metal.     In  the  first  case  the  steel  aids  by  carrying  a 
part  of  the  load  directly,  the  stresses  in  the  two   materials 
being  proportional  to  their  moduli  of  elasticity.     In  the  other 
case  the  steel  supports  the  concrete  laterally,  preventing  lateral 
expansion  to  a  greater  or  less  degree,  and  thus  strengthening 
the  concrete.     Usually  both  methods  are  more   or  less  com- 
bined, the  longitudinal  rods  being  frequently  bound  together 
at  intervals  by  circumferential  bands  of  some  sort,  and  on 
the  other  hand  hoops  or  spiral  wire  being  conveniently  held 
in  place  by  longitudinal  rods.      Experiments  show  that  both 
types  of  reinforcement  are  effective  in  raising  the  ultimate 
strength  of  a  column,  but  conclusive  results  have  not  been 
reached  as  to  the  true  relative  effect  of  different  types  and 
amounts  of  reinforcement. 

95.  Columns  with  Longitudinal  Reinforcement. — As  long  as 
the  steel  and  concrete  adhere  the  relative  intensities  of  stress 
in  the  two  materials  will  be  as  their  moduli  of  elasticity,  using 
the  modulus  as  explained  in  Art.  24. 

Let  A  denote  total  cross-section  of  column; 
Ac  cross-section  of  concrete; 

As     "      cross-section  of  steel; 
p       "      ratio  of  steel  area  to  total  &rea,=A8/A', 
fc       "     stress  in  concrete; 
n  ratio  of  moduli  of  steel  and  concrete  at  the 

given  stress  fc,  =  Es/Ec\ 

P  total  strength  of  a  plain  column  for  the  stress  /c; 

Pf      "     total  strength  of  a  reinforced  column  for  the 
stress  fc. 


§95.]  STRENGTH  OF  COLUMNS.  107 

Then  P  =  /<4    ........     (a) 


and  P'  =  fcAc+fsAa=fc(A 

whence  Pf  =  fcA[l  +  (n-l)p],    ........     (1) 


from  which  also 

.....    (2) 


The  relative  increase  in  strength  caused  by  the  reinforce- 
ment is 

P'-P 
-p-=(n-l)p (3) 


The  elastic  limit  of  the  steel,  if  low,  may  affect  the  ulti- 
mate strength  of  the  column.    The  value  of  P'  is  then 


(4) 


in  which  fa  is  the   elastic   limit  strength   of  the  steel.     (See 
Chapter  IV  for  further  discussion  of  this  question.) 

Eq.  (2)  is  convenient  to  use  in  determining  the  relative- 
strength  of  a  reinforced  as  compared  to  a  plain  concrete  col- 
umn for  a  given  percentage  of  steel.  Thus  if  p  =  l%  and 

P' 
n  =  15,  we  have  p-  =  1  +  0.  14  =  1.14.     Thus   a  reinforcement  of 

1%  increases  the  strength  by  14%. 

From  these  relations  it  is  seen  that  the  relative  increase 
in  strength  caused  by  a  given  amount  of  reinforcement  depends 
on  the  value  of  n  and  is  greater  the  larger  n  is. 

The  economy  of  steel  reinforcement  is  also  dependent 
upon  the  working  stresses  permissible  in  the  concrete  since 


108 


GENERAL  THEORY, 


fiCH.  III. 


/«=tt/c.  The  following  table  shows  the  various  working  stresses 
in  the  steel  corresponding  to  various  values  of  working  stress 
in  the  concrete  and  to  various  values  of  the  modulus  Ec,  there 
is  given  also  the  percentage  increase  in  strength  for  each  one 
per  cent  of  steel. 


TABLE  No.  6. 
LONGITUDINAL  REINFORCEMENT  OF  COLUMNS. 


lbsVin.2 

EC, 

lbs/in2 

Ratio  of  Moduli, 
n 

!» 

Ibs/in2 

Percentage  Increase 
in  Strength  for 
each  1  %  Rein- 
forcement. 

f 

750,000 

40 

12,000 

39 

Qfin 

1,000,000 

30 

9,000 

29 

oUU 

1,500,000 

20 

6,000 

19 

I 

2,000,000 

15 

4,500 

14 

r 

1,000,000 

30 

12,000 

29 

40fi 

1,500,000 

20 

18,000 

19 

T;UU                "S 

2,000,000 

15 

6,000 

14 

I 

2,500,000 

12 

4,800 

11 

r 

1,000,000 

30 

15,000 

29 

500 

1,500,000 
2,000,000 

20 
15 

10,000 
7,500 

19 
14 

I 

2,500,000 

12 

6,000 

11 

f 

1,500,000 

20 

15,000 

19 

fiftfl 

2,000,000 

15 

10,000 

14 

OUU               "\ 

2,500,000 

12 

7,200 

11 

[ 

3,000,000 

10 

6,000 

9 

f 

2,000,000 

15 

12,000 

14 

son 

2,500,000 

12 

9,600 

11 

oUU             •< 

3,000,000 

10 

8,000 

9 

I 

3,500,000 

8.6 

6,900 

7.6 

From  this  table  the  relation  among  the  various  quantities 
may  be  clearly  appreciated.  It  is  to  be  noted  that  the  work- 
ing stresses  in  the  steel  must  be  relatively  low  except  in  the 
unusual  combination  of  high  working  stresses  in  the  concrete 
with  low  modulus.  High-grade  concrete,  permitting  high  work- 
ing stresses,  will  have  a  high  modulus.  For  further  discussion 
of  the  relations  of  working  stresses,  see  Chapter  V. 


§96,]  STRENGTH  OF  COLUMNS.  109 

Examples.  —  (1)  What  will  be  the  safe  strength  of  a  column  15"X15" 
in  cross-section  which  is  reinforced  with  1.5%  of  steel,  the  working  stress 
in  the  concrete  being  400  lbs/in2.  Take  n  =  15. 

From  eq.  (1)  we,  have 

P'  =  400X  15X15  X  (l  +  14xi5)  =90,000(1  +0.21)  =  108,900  Ibs. 


The  strength  of  the  plain  concrete  column  would  be  90,000  Ibs.,  and 
the  relative  increase  in  strength  is  21%.  The  stress  in  the  steel  would 
be  15X400  =  6000  lbs/in2. 

(2)  The  area  of  a  column  is  120  sq.  in.,  load  to  be  carried  is  60,000 
Ibs.,  and  working  stress  on  the  concrete  is  400  lbs/in2.  What  percentage 
•of  steel  will  be  required?  Take  n  =  15. 

The  safe  strength  of  a  plain  concrete  column  would  be  120X400  = 

P'    60 
48,000  Ibs.    Hence,  from  eq.  (2),  =-  =  —  =  1  +  (15-l)p.    Hence 


96.  Columns  with  Hooped  Reinforcement.  —  Whenever  a 
material  subjected  to  compression  in  one  direction  is  restrained 
laterally,  then  lateral  compressive  stresses  are  developed  which 
tend  to  neutralize  the  effect  of  the  principal  compressive  stresses 
and  thus  to  increase  the  resistance  to  rupture.  Were  the  com- 
pressive stresses  equal  in  all  directions  there  would  be  no  rup- 
ture (as  there  would  be  no  shear).  The  strengthening  effect 
of  lateral  banding  depends  then  upon  the  rigidity  of  the 
bands,  that  is,  upon  the  amount  of  steel  used  and  its  closeness 
of  spacing.  Its  elastic  limit  may  also  affect  the  ultimate 
strength  of  the  column. 

On  the  basis  of  the  relative  lateral  and  horizontal  defor- 
mation of  the  concrete  (Poisson's  ratio)  it  is  possible  to  deduce 
a  theoretical  relation  between  the  lateral  and  the  longitudinal 
stresses,  and  thence  the  portion  of  the  longitudinal  stress 
remaining  unbalanced.  Let  /*  =  Poisson's  ratio,  /c= unbal- 
anced or  excess  of  longitudinal  over  lateral  compressive  unit 
stress,  //  =  total  longitudinal  unit  stress,  fs  =  unit  tensile  stress 


110  GENERAL  THEORY.  [Cn.  III. 

in  steel,  p  =  steel  ratio  (reinforcement  to  be  closely  spaced). 
We  find  approximately 


and 

f.  =  pnfe*.      .     .     .     .....     (2) 

Hecent  experiments  by  Talbot  indicate  that  Poisson's  ratio 
for  concrete  is  quite  small,  probably  not  greater  than  T<T  or  J. 

*  Demonstration.  (See  Johnson's  "Materials  of  Construction".)  —  Let 
H=  Poisson's  ratio;  p=  steel  ratio  considered  as  a  thin  cylinder  of  equivalent 
area  surrounding  the  concrete;  As  =  cross-section  of  this  steel  cylinder; 
r=  radius.  Then 


A8=  pnr2     and     thickness  of  cylinder  =  ££—  =  p—. 

ZflT          2i 

With  no  steel  banding  the  stress  /</  would  cause  a  proportionate  lateral 

t  / 
swelling  of  «r/t.     If  the  actual  stress  in  the  steel  is  /8  then  the  compression  per 

sq.  in.  developed  in  the  concrete  by  the  steel  reinforcement  =  f8p^--t-r  ='-?£• 

.  2i  2i 

This  compression  caused  by  the  banding  is  equal  in  all  horizontal  directions, 
and  has  the  same  effect  on  distortion  as  two  pairs  of  equal  compressive  forces 
acting  on  two  sets  of  faces  of  a  cube.  The  resultant  lateral  compression  due 

to  these  horizontal  forces  is  equal  to  j~  (1  —  //).  Combining  this  compression 
with  the  lateral  swelling  caused  by  /</  we  have  the  net  lateral  deformation 
equal  to  ~u  —  ~K-(^-  —  j«).  This  net  deformation  must  equal  the  actual 

&c         £&c 

deformation  in  the  steel  under  the  stress  f8,  which  is  ~  or  ~~.     Hence  we 

J^a       nhic 

have 

/''        f*P(l     wW  /• 

7^  ~~  OF"  U        /*  '  --  jjr"  • 

Jzic  •-,    z&c  nHic 

A  part  of  /</  may  be  considered  to1  be  balanced  by  the  lateral  compression 
of  -—-;    it  is  the  unbalanced  portion  only  which  is  significant.     Call  this 

unbalanced  portion  /c;  then  fc'=fc+  '-^.  Then  eliminating  fa  from  these 
two  equations  We  find  for  /c'  the  value 


§96.]  STRENGTH  OF  COLUMNS.  Ill 

At  the  latter  value  eqs.  1)  and  (2)  would  become  //=/«. 
(1  +  np/16),  and  fs  =  Jn/c.  Comparing  these  equations  with  those 
of  Art.  95  it  would  appear  that  within  the  limit  of  elasticity  the 
hooped  reinforcement  is  much  less  effective  than  longitudinal 
reinforcement ;  in  fact  it  would  seem  that  very  little  stress  can 
be  developed  in  the  steel  under  elastic  conditions  as  here 
assumed.  Such  reinforcement  may,  however,  be  quite  effective- 
in  increasing  the  ultimate  strength  of  a  column. 

Results  of  tests  appear  to  accord  in  a  general  way  with 
these  theoretical  relations.  Hooped  columns  have  a  relatively 
large  deformation,  reaching  at  an  early  stage  a  deformation 
equal  to  the  maximum  for  plain  concrete.  Under  further 
loading  the  concrete  is  prevented  by  the  banding  from  actual 
failure,  but  continues  to  compress  and  to  expand  laterally, 
increasing  the  tension  in  the  bands,  the  elasticity  of  the  bands 
rendering  the  column  in  large  degree  still  elastic.  Final  failure 
occurs  upon  the  breakage  of  the  bands  or  their  excessive 
stretching.  Banded  columns  thus  exhibit  a  toughness  or 
ductility  much  greater  than  other  forms,  but  without  a  cor- 
responding increase  in  stiffness  under  lower  loads.  Ultimate 
failure  is  likely  to  be  long  postponed  after  the  first  signs  of 
rupture,  and  the  column  will  sustain  greatly  increased  loads 
even  after  the  entire  failure  of  the  shell  of  concrete  outside 
the  bands. 

Consid£re  has  made  extensive  theoretical  and  experimental 
investigations  of  hooped  columns,  from  which  he  concludes  that 

We  also  have 


For  ordinary  values  of  p  eqs.  (a)  and  (6)  are  reduced  approximately  to 

..........     (!) 


and 

/.-/m/c  .......  .    .     (2) 


112  GENERAL  THEORY.  [On.  III. 

the  ultimate  strength  is  given  by  the  formula 

P'=fcA+2AfspA, (5) 

in  which  fc  is  the  strength  of  concrete  and  f8  is  the  elastic-limit 
strength  of  the  steel.  This  formula  virtually  counts  the  steel 
worth  2.4  times  as  much  as  in  longitudinal  reinforcement. 
(For  further  discussion  see  Chapter  IV.) 


CHAPTER  IV. 
TESTS   OP  BEAMS   AND  COLUMNS. 

BEAMS. 

97.  Methods  of  Failure  of  a  Reniforced-concrete  Beam. — 

A  reinforced-concrete  beam  tested  to  destruction  will  usually 
fail  in  one  of  three  ways: 

(a)  By  the  yielding  of  the  steel  at  or  near  the  section 
of  maximum  bending  moment. 

(6)  By  the  crushing  of  the  concrete  at  the  same  place. 

(c)   By  a  diagonal  tension  failure  of  the  concrete  at  a 

place  where  the  shear  is  large. 

Methods  (a)  and  (6)  may  be  called  "moment"  failures.  Method 
(c)  is  sometimes  called  a  shear  failure,  but  this  term  is  some- 
what misleading,  as  the  concrete  in  such  cases  does  not  fail  by 
shearing. 

(a)  As  a  beam  is  progressively  loaded  and  the  steel  has 
reached  its  yield  point  any  further  load  will  rapidly  increase 
the  deformation.  ;  The  effect  of  this  is  to  open  up  large  cracks 
in  the  tension  side  and  to  raise  the  neutral  axis.  This  causes 
a  rapid  increase  in  the  compressive  stress  in  the  concrete  and 
ultimate  failure  soon  occurs  by  the  concrete  crushing.''  Such 
yielding  may  also  result  in  final  failure  by  diagonal  tension 
if  large  shear  exists  near  the  place  of  maximum  moment.  In 
this  case  the  primary  cause  of  failure  is  the  yielding  of  the  steel 
and  such  failure  may  properly  be  called  a  tension  failure.  The 
additional  loa$  carried  after  the  yield  point  is  reached  depends 
on  the  excess  strength  of  the  concrete,  position  of  loads,  and 

113 


114  TESTS  OF  BEAMS  AND  COLUMNS.  [Cn.  IV. 

other  causes,  but  it  is  usually  not  large  and  cannot  be  safely 
considered.  The  yield  point  of  the  steel  may  therefore  be  con- 
sidered its  ultimate  strength  for  reinforcing  purposes. 

(b)  'If  the  beam  is  relatively  long  and  the  amount  of  steel 
is  sufficient  so  that  the  crushing  strength  of  the  concrete  is 
reached  before  the  yield  point  of  the  steel,  a  failure  by  crushing 
is  likely  to  result.  In  this  case  tension  cracks  may  appear,  but 
will  not  become  large.  Fig.  38,  (a)  and  (b),  illustrates  methods 
of  failure  (a)  and  (b)  respectively. 

i i 


/      /       /  \ 


(&) 


(c) 
FIG.  38.— Methods  of  Failure  of  Beams. 


(c)  Diagonal  tension  failures  are  likely  to  occur  whenever 
large  shearing  stresses  exist  together  with  considerable  hori- 
zontal or  moment  stresses,  and  when  no  special  provision  is 
made  for  such  conditions.  This  is  especially  likely  to  occur 
in  beams  of  relatively  great  depth,  beams  having  a  ratio  of 
depth  to  length  of  more  than  about  1 : 10  being  likely  to  fail 
in  this  way  if  no  special  provision  is  made  for  web  rein- 
forcement. 

Fig.  38,  (c),  illustrates  the  typical  diagonal  tension  failure 
where  only  horizontal  bars  are  used.  The  initial  crack  forms 
at  a.  This  gradually  extends  upwards  in  an  inclined  line  and 


UNIVERSITY 

OF 


§99.]  METHODS  OF  FAILURE  OF  BEAMS.  119 

a  little  later  the  concrete  begins  to  fail  in  a  horizontal  tension 
crack  just  above  the  rods,  progressing  from  a  towards  the  end 
of  the  beam.  Tension  along  this  line  is  brought  about  by  the 
new  conditions  existing  after  the  concrete  has  become  cracked 
along  the  diagonal  line  and  the  normal  diagonal  tension  has 
thus  ceased  to  act.  Usually  this  horizontal  crack  rapidly  ex- 
tends to  the  end  of  the  beam  and  the  failure  is  complete.  In 
other  cases  the  diagonal  crack  may  extend  to  the  top  of  the 
beam,  allowing  the  part  on  the  right  to  drop  down  and  causing 
final  failure.  In  such  a  case  the  concrete  on  the  left  may  remain 
intact.  Figs.  39  and  40  are  photographs  representing  "  diagonal- 
tension  "failures. 

A  rupture  of  the  concrete  on  a  diagonal  line  also  causes 
an  increase  in  the  stress  on  the  rod  at  a,  as  shown  more  fully  in 
Art.  108.  This  may  result  in  a  failure  of  bond,  especially  if 
the  support  is  too  near  the  end  of  the  beam. 

Final  failure  thus  often  results  from  stresses  which  are  devel- 
oped after  initial  failure  has  occurred,  and  while  the  cause 
of  final  failure  is  important  from  the  standpoint  of  uHimate 
strength,  yet  of  more  importance  in  design  is  the  initial  failure 
and  its  cause.  Other  conditions  besides  those  already  men- 
tioned may  influence  final  failure  so  as  often  to  mislead  the 
observer  as  to  the  cause  of  the  initial  failure. 

98.  Minor  Causes  of  Failure. — Slipping  of  the  bars  may 
cause  failure,  but  under  usual  conditions  it  will  not  occur; 
and  as  it  can  readily  be  obviated  by  proper  construction  it  need 
not  be  considered  as  limiting  the  strength  of  the  beam.  Failure 
by  the  shearing  of  the  concrete  near  the  support  is  possible  where 
the  load  is  very  close  thereto,  but  as  the  shearing  strength  of 
concrete  is  about  one-half  the  crushing  strength,  such  failures 
are  exceedingly  unlikely  and  need  rarely  be  considered.  The 
usual  so-called  "shear"  failures  are  in  reality  diagonal-tension 
failures. 

99-  Tests  of  Beams  Giving  Steel-tension  Failures.— The 
diagrams  of  Figs.  41  and  42  present  in  a  roughly  classified  form 
results  of  the  most  important  tests  on  reinforced-concrete  beams 


120  TESTS  OF  BEAMS  AND  COLUMNS.  [Cn.  IV. 

in  which  the  failure  appears  to  have  been  caused  primarily  by 
the  yielding  of  the  steel.  In  such  a  case  the  strength  of  the 
beam  is  directly  proportional  to  the  elastic-limit  strength  of 
the  steel,  and  hence  the  tests  have  been  classified  as  nearly  as 
practicable  with  respect  to  this  limit.  The  tests  are  thus 
divided  into  four  groups  according  to  values  for  the  elastic 
limit  as  given  in  the  diagrams.  On  each  of  the  diagrams  are 
drawn  theoretical  curves  of  strength  using  values  for  the  steel 
stress  corresponding  to  the  elastic  limit  for  the  group.  The 
full  line  is  based  upon  the  parabolic  law  of  stress  variation,  the 
full  parabola  being  used;  the  dotted  line  is  based  upon  the 
straight-line  law  of  stress  variation.  The  value  of  n  was  taken 
at  15.* 

Considering  the  nature  of  the  material  and  of  the  tests  the 
agreement  between  theory  and  experimental  results  is  very  satis- 
factory. It  is  to  be  expected  that  the  theoretical  values 
should  represent  minimum  rather  than  average  results,  since 
the  strength  of  a  beam  as  determined  by  the  elastic  limit  of  the 
steel  should  be  at  least  equaled,  and  generally  slightly  exceeded, 
in  a  test  if  failure  does  not  occur  in  some  other  way.  If  the 
conditions  are  favorable  the  strength  may  considerably  exceed 
that  corresponding  to  the  elastic  limit  of  the  steel,  and  in  a 
few  tests  the  steel  has  been  pulled  apart  before  complete  col- 
lapse has  taken  place.  Such  excess  of  strength  cannot  be 
counted  upon,  however,  as  is  well  indicated  in  the  diagrams. 


*  The  sources  of  information  are  as  follows: 

1.  Boston  Transit  Commission,  Fourth  Annual  Report,  1904. 

2.  Bulletins  Nos.  1  and  4,  University  of  Illinois,  Engineering  Ex- 

periment Station 

3.  Jour.  West.  Soc.  Eng.,  Vol.  X,  1905,  p.  705  (C.,  M.  &  St.  P.  R'y 

Co.'s  tests). 

4.  Jour.  West,  Soc.  Eng.,  Vol.  IX,  1904,  p.  239  (tests  of  M.  A.  Howe). 

5.  Bulletins  Nos.  4  and  6,  Engineering  Series,  University  of  Wis- 

consin, 1907. 

6.  Proc.  Am.  Soc.  Test.  Materials,  Vol.  IV,  1904,  p.  508  (Univ.  of 

Penn.  tests). 

7.  Eng.  Record,  Vol.  LI,  1905,  p.  545  (Purdue  Univ.  tests). 


§99. 
1000 

900 
800 
700 
600 
500 
400 
300 
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1100 
1.000 
900 
300 
700 

600 
500 
400 
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200 
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]                                     TESTS  OF  BEAMS.                                       121 

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Percentage  Beinfor.c.ement 
FIG.  41. — Steel-tenison  Failures. 


2.0 


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1200 
1100 
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TESTS  OF  BEAMS  AND  COLUMNS.                [Cn.  IV 

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FIG.  42. — Steel-tension  Failures. 


§  100.]  TESTS  OF  BEAMS.  123 

In  the  tests  of  the  Boston  Transit  Commission,  which  range 
uniformly  high,  the  conditions  were  favorable,  inasmuch  as 
the  beams  were  tested  with  center  load.  The  concrete  was 
also  of  very  high  grade,  having  a  crushing  strength  of  about 
4000  lbs/in2,  thus  enabling  the  steel  to  elongate  very  con- 
siderably before  final  failure  occurred  through  the  crushing 
of  the  concrete. 

No  distinction  has  been  made  in  these  diagrams  between  the 
different  grades  of  concrete  employed.  Variations  in  concrete 
will  affect  the  results  only  by  slightly  affecting  the  position 
of  the  neutral  axis,  and  hence  the  resisting  moment  of  the 
steel,  and  by  postponing  somewhat  the  final  failure,  as  noted 
above. 

100.  Results  from  Individual  Tests. — In  many  of  the  tests 
made  at  the  University  of  Illinois  and  at  the  University  of 
Wisconsin,  and  in  the  tests  made  by  Professor  Howe,  exten- 
someters  were  used  to  measure  distortions  so  that  the  deforma- 
tion of  the  steel  and  of  the  extreme  fiber  of  the  concrete  could 
be  calculated  and  the  neutral  axis  determined.  Typical  results 
are  shown  in  Figs.  43  and  44.  In  Fig.  43  the  porportions  were 
such  that  the  failure  occurred  by  diagonal  tension;  neither 
the  steel  nor  the  concrete  was  stressed  to  the  limit  of  failure. 
During  the  first  stage  of  the  test,  up  to  a  load  of  about  2500 
pounds,  the  deformations  in  both  steel  and  concrete  are  pro- 
portional to  the  loads.  Up  to  this  point  the  tension  defor- 
mation has  not  been  great  enough  to  begin  to  rupture  the 
concrete,  but  with  increasing  loads  and  deformations  the  con- 
crete begins  to  fail,  as  shown  by  the  appearance  of  minute 
cracks  (the  "water-marks"  discussed  in  Art.  42),  indicated  on 
the  diagram  by  the  letters  W.M.  The  deformation  at  the 
first  "water-mark"  in  this  case  was  about  .00018,  correspond- 
ing to  a  stress  of  270  lbs/in2,  assuming  a  modulus  of  elasticity 
of  1,500,000.  The  first  visible  crack  appeared  at  the  point 
marked  C. 

The  failure  of  the  concrete  in  tension  takes  place  somewhat 
gradually  and  causes  a  gradual  increase  in  the  rate  of  deforma- 


124 


TESTS  OF  BEAMS  AND  COLUMNS. 


[Cn,  IV. 


.Deflection  in  Inches 
01      02     03     Ol4     015     016     07     08     09      10 


.0004  .0008          .0012      -    .0016  .0020  .0034 

Deformation  per  Unit  Length 

FIG.  43. 


.0038 


Deflection  in  Inches 
02      Ol3      OJ4     OJ5      Oi6      0 


.0008  .0012      -    .0016  .0020 

Deformation  per  Unit  Length 


FIG.  44. 


§  101.]  POSITION  OF  NEUTRAL  AXIS.  125 

tion  as  indicated  by  the  curved  part  of  the  diagram  between 
loads  of  2500  and  4000  pounds.  After  the  concrete  has  ceased 
to  offer  any  considerable  resistance  in  tension  the  deformations 
again  become  nearly  proportional  to  the  loads,  but  at  a  different 
ratio  from  that  obtaining  previously,  giving  nearly  straight 
lines  for  both  steel  and  concrete — in  this  case  to  the  end  of  the 
test. 

In  Fig.  44  the  amount  of  steel  was  small  and  a  tension 
failure  occurred.  This  is  indicated  by  the  great  deformations 
at  the  end  of  the  test.  The  curves  in  the  early  stages  of  the  test 
are  very  similar,  in  general  form,  to  those  in  Fig.  43. 

In  the  case  of  a  compressive  failure  the  curve  for  compres- 
sion shows  an  increased  rate  of  deformation  towards  the  end, 
somewhat  similar  to  the  diagram  for  simple  compression. 
t/  101.  Position  of  Neutral  Axis  and  Value  of  n. — In  Figs.  45 
and  46  are  plotted  the  results  of  experiments  in  which  the 
position  of  the  neutral  axis  has  been  determined.  The  position 
is  given  for  three  stages  of  the  tests,  at  one-fourth,  one-half, 
and  three-fourths  of  the  ultimate  load.  On  the  diagrams  are 
plotted  the  theoretical  positions  of  the  neutral  axis  for  various 
values  of  n.  The  full  lines  are  based  on  the  straight-line  stress 
variation  assumption,  and  the  dotted  lines  on  the  assumption 
of  a  parabolic  law  in  accordance  with  Professor  Talbot's  method 
(see  Art.  65).  The  dotted  lines  have  been  drawn  only  for  a 
single  value  of  15  for  n.  For  the  three-quarter  load  the  dotted 
line  for  n  =  15  would  coincide  very  closely  with  the  full  line 
for  n  =  20.  The  value  of  q  has  been  taken  at  J,  J,  and  f ,  respec- 
tively. 

It  will  be  noted  that  for  the  one-quarter  loads  and  the  small 
percentages  of  steel  the  neutral  axis  is  more  uncertain  and 
generally  lower  than  for  the  higher  loads  and  larger  percentages. 
This  is  due  doubtless  to  the  relatively  large  influence  of  the 
tensile  strength  of  the  concrete  in  such  cases.  From  these  re- 
sults it  would  seem  that  a  value  of  15  for  n  is  as  low  as  would  be 
warranted,  even  for  the  quarter  load,  which  is  not  far  from  the 
usual  safe  load.  This  corresponds  to  a  value  of  Ec  of  2,000,000, 


126 


TESTS  OF  BEAMS  AND  COLUMNS. 


[Cn.  IV. 


At  One-fourth  Ultimate  Load 
1.0  1.5 


2.0 


0.2 
0.1 
0.0 
0.8 
1.0 
0 
0.2 


L 

I 

2 

PLnO.8 


1.0 


0.2 


0.4 


0.0 


0.8 


1.0 


At  One-half  Ultimate  Load 


At  Three-fourths  Ultimate  Load 


0.5 


1.0  1.5 

Percentage  Reinforcement 


2.0 


FIG.  45. — Position  of  Neutral  Axis.     (1:2:5  Concrete.) 


§  101.] 


POSITION  OF  NEUTRAL  AXIS. 


127 


At  One-fourth  Ultimate  Load 


At  One-half  Ultimate  Load 


At  Three-fourths  Ultimate  Load 


1.0  1.5  2.0 

Percentage  Reinforcement 

FIG.  46.— Position  of  Neutral  Axis.     (1:3:6  Concrete.) 


128  TESTS  OF  BEAMS  AND  COLUMNS.  [Cn.  IV. 

which  is  somewhat  low  as  determined  by  compressive  tests. 
A  value  of  ?i  =  10,  corresponding  to  #  =  3,000,000,  does  not 
accord  with  results  from  bending  tests.  If  the  comparison 
between  measured  and  calculated  positions  of  the  neutral  axis 
be  made  on  the  basis  of  the  parabolic  law  of  stress  variation  the 
results  will  differ  considerably  in  the  latter  stages  of  the  tests, 
but  very  slightly  at  the  quarter  load.  It  should  be  noted, 
however,  that  it  is  only  with  the  high  percentages  of  steel  that 
the  concrete  stress  reaches  nearly  to  its  ultimate  value,  and 
hence  is  the  only  condition  where  the  full  parabolic  law  can  be 
expected  to  give  consistent  and  rational  results. 

In  some  of  the  tests  whose  results  are  plotted  here  the  con-« 
crete  was  cut  away  from  the  steel  for  the  measured  distance, 
leaving  it  exposed.    The  position  of  the  neutral  axis  was  very 
slightly  affected. 

1 02.  Observed  and  Calculated  Stresses  in  Steel. — Where 
the  neutral  axis  is  determined  by  extensometer  measurements 
a  check  upon  theoretical  results  can  be  obtained  by  calculating 
the  stress  in  the  steel  in  two  ways:  (1)  from  the  observed 
deformations  at  the  plane  of  the  steel,  and  (2)  from  the  known 
bending  moment  and  known  position  of  the  neutral  axis.  In 
the  first  calculation  the  tensile  strength  of  the  concrete,  which 
is  neglected,  causes  some  error,  especially  under  light  loads, 
and  in  the  second  calculation  the  exact  position  of  the  centroid 
of  pressure  in  the  concrete,  especially  in  the  later  stages  of  the 
test,  is  to  a  small  degree  uncertain,  but  as  the  variation  in  • 
steel  stress  is  only  about  2%,  using  the  two  extreme  assump- 
tions of  stress  variation,  this  source  of  error  is  not  great.  Table 
No.  7  presents  several  representative  results  derived  from  such 
calculations.  The  stresses  calculated  from  moments  are  based 
on  the  assumption  that  the  concrete  takes  no  tension. 

Tests  have  been  made  at  the  University  of  Illinois  and  at 
the  University  of  Wisconsin  in  which  the  rods  have  been  exposed 
for  a  considerable  distance  along  the  center  of  the  beam,  and 
thus  have  been  much  less  affected  by  any  possible  tensile 
stress  in  the  concrete.  Measurements  of  extension  made  in  such 


§  103.] 


POSITION  OF  NEUTRAL  AXIS. 


129 


cases  show  little  variation  from  those  made  on  the  ordinary 
beam. 

TABLE  No.  7. 
STRESSES  IN   STEEL   REINFORCEMENT. 


Calculated  Stress  in  Steel, 

Observed 

lbs/in2. 

Authority. 

Per  Cent 
Reinforcement  . 

Position  of 
Neutral  Axis, 

k. 

From 

From  Exten- 

Moments. 

sions  in  Steel. 

.74 

.410 

33,100' 

36,000 

Talbot; 
Bull.    Univ.    of 
III,  1906. 

1.23 
1.60 
1.66 
1.84 

.470 
.501 
.505 
.606 

35,000 
29,500 
30,600 
25,600 

36,000 
35,400 
30,000 
27,200 

1.84 

.552 

28,300 

30,000 

Withey; 
Bull.     Univ.    of 
Wis.,  1907. 

2.9 
2.9 

.670 
.6*  0 

3^,200 
31,600 

36,000 
33,000 

Considering  the  nature  of  such  experiments  the  results 
obtained  may  be  considered  as  according  with  theory  very  sat- 
isfactorily. 

y  103.  Compressive  Stresses  in  Concrete  in  Beams  and 
in  Compression  Specimens. — An  important  question  relating 
to  proper  working  stresses  is  whether  the  ultimate  compressive 
strength  of  concrete  in  a  beam  is  the  same  as  determined  by  a 
direct  compression  test. 

The  results  of  certain  tests  indicate  that  the  compressive 
strength  and  ultimate  deformation  in  a  beam  may  be  some- 
what greater  than  in  a  prismatic  compressive  piece;  and  iu 
would  seem  that  the  differences  in  condition  are  sufficient  to 
make  such  a  difference  possible.  In  a  compressive  specimen 
the  material  is  free  to  shear  in  any  direction,  thus  limiting 
the  strength  of  the  specimen  to  its  weakest  shearing  plane.  In 
a  beam  the  (shear)  failure  is  practically  confined  to  planes 
perpendicular  to  the  side  of  the  beam.  Furthermore,  in  a 
beam  the  material  is  not  subjected  to  the  secondary  stresses 
due  to  possible  poor  bedding  of  the*  test  specimen  or  non- 


130 


TESTS  OF  BEAMS  AND  COLUMNS. 


[On.  IV. 


parallel  motion  of  the  testing  machine,  as  is  the  case  in  compres- 
sion tests. 

In  most  of  the  tests  reported  both  the  beams  and  the  ac- 
companying compression  specimens  have  been  hardened  in 
air.  Under  these  conditions  there  is  usually  some  drying-out 
effect  resu  ting  in  a  weaker  concrete  than  if  hardened  in  water, 
and  owing  to  the  smaller  dimensions  of  the  compressive  speci- 
mens the  effect  will  be  greater  with  them  than  with  the  rela- 
tively large  beams.  Many  tests  have  therefore  shown  a  com- 
pressive strength  of  concrete  in  the  beam  considerably  greater 
than  results  obtained  on  cubes.  When  both  beam  and  cube 
are  hardened  in  water  the  results  do  not  differ  greatly.  The 
following  are  some  results  obtained  on  tests  made  relative 
to  this  point.*  The  beams  were  8"XlO"  net  section  and 
12  ft.  span.  They  were  reinforced  with  2J%  of  steel  and 
gave  compressive  failures.  The  cubes  were  4  inches  in  dimen- 
sion and  the  cylinders  6"  in  diameter  by  18"  high. 


Stress  in  Concrete  at  Rupture,  lbs/in2. 

Beam. 

Cube. 

Cylinder. 

Hardened 
Hardened 

ri.. 

1770 
1460 

1810 
1850 

1187 
1350 

1450 
1750 

1380 
1295 

1265 
1680 

m  air         <  2 

[3 

in   water  <.  * 

The  stresses  in  the  beams  were  calculated  on  the  basis  of  the 
parabolic  variation  of  stress,  the  neutral  axis  being  determined 
by  extensometers. 

It  will  be  seen  that  in  case  of  the  specimens  hardened  in 
air  there  is  a  marked  difference  in  strength,  but  where  hardened 
in  water  the  difference  is  much  less.  The  difference  is  hardly 
sufficient  to  warrant  much  consideration  in  the  determination 
of  working  stresses. 


*  Bulletin  No.  6,  Engineering  Series,  University  of  Wisconsin,  1907. 


J105.]  POSITION    OF   NEUTRAL  AXIS.  131 

V    /i 04.  Conclusions    Regarding    Moment    Calculations. — 

The  comparison  of  experimental  results  with  theoretical  analysis 
herein  given  shows  that  the  simple  beam  theory  as  generally 
employed,  neglecting  the  tension  in  the  concrete,  can  be  used 
with  confidence.  In  particular,  the  results  appear  to  show 
that  calculated  on  the  basis  of  such  theory  the  yield  point 
(commonly  called  the  elastic  limit)  of  the  steel  may  safely  be 
taken  as  its  ultimate  strength  in  reinforced  beams;;  that  the 
crushing  strength  of  concrete  as  determined  by  tests  on  cubes 
hardened  under  exactly  similar  conditions  as  the  beams  will  be 
fully  realized  in  the  beam;  that  for  working  loads  the  straight- 
line  law  of  stress  variation  is  sufficiently  exact;  that  the  value 
of  n  may  be  taken  at  about  15,  but  that  great  accuracy  in  this 
respect  is  unnecessary;  that  for  ultimate  values,  especially 
where  the  concrete  is  near  failure,  the  parabolic  assumption 
of  stress  variation  may  well  be  used. 

105.  Tests  in  which  Failure  Occurred  by  Diagonal  Ten- 
sion. Influences  Affecting  Failure  by  Diagonal  Tension. — The 
strength  of  a  beam  in  diagonal  tension  is  not  a  simple  function 
of  the  shear,  but  as  shown  in  Art.  90  it  depends  also  upon  the 
horizontal  tension  or  bending-moment  stresses  in  the  concrete. 
These  will  in  turn  depend  upon  the  actual  bending  moment 
at  the  section  of  failure  and  the  amount  of  horizontal  reinforce- 
ment, a  large  percentage  of  reinforcement  reducing  the  horizontal 
deformation  and  therefore  the  tension  in  the  concrete  and 
tending  to  strengthen  the  beam  as  regards  failure  in  diagonal 
tension.  The  strength  of  the  beam  therefore  depends  upon  the 
relation  between  shear  and  bending  moment  and  upon  the 
amount  of  reinforcement.  The  chief  factor  is,  however,  the 
shearing  stress. 

From  the  preceding  considerations  it  is  evident  that  the 
nature  of  the  loading  will  influence  the  strength  of  the  beam. 
Most  structures  are  calculated  for  uniform  or  approximately 
uniform  loading,  and  in  experimental  work  two  concentrated 
loads  applied  at  the  third  points  are  commonly  used  as  repre- 
senting roughly  the  conditions  which  exist  in  the  uniformly 


132 


TESTS   OF  BEAMS  AND   COLUMNS. 


[Cn.  IV. 


loaded  beam.  Fig.  47  represents  the  variation  in  moment 
and  shear  in  a  beam  loaded  at  the  third  points,  while  Fig.  48 
shows  similar  curves  for  a  uniformly  loaded  beam.  It  is 
to  be  noted  that  in  the  first  case  maximum  shear  occurs  where 
maximum  moment  exists,  while  in  the  latter  case  maximum 
shear  occurs  at  the  point  of  zero  moment.  In  the  former  case 


Shear 


FIG.  47.  FIG.  48. 

diagonal-tension  failure  will  occur  just  outside  the  loads,  while 
in  the  latter  case  it  will  occur  nearer  to  the  support  where  the 
moment  is  considerably  less  than  the  maximum.  Conditions 
as  to  shear  are  therefore  somewhat  more  favorable  in  the  con- 
tinuously loaded  beam.  A  single  concentrated  load  causes  less 
shear  for  a  given  moment  than  the  double  load,  and  is  there- 
fore more  favorable  as  regards  shear. 

As  continuous  beams  are  commonly  used  in  building  con- 
struction it  will  be  useful  to  note  here  the  variation  in  shear 
and  moment  in  such  a  beam.  This  is  shown  in  Fig.  49,  and  it 
will  be  seen  that  the  conditions  here  are,  quite  unfavorable, 
large  shear  occurring  near  the  supports  where  the  negative 
bending  moment  is  large. 

Whether  a  beam  will  fail  from  moment  stresses  or  shearing 
stresses  will  depend  largely  upon  its  relative  length  and  depth. 
For  any  given  distribution  of  loads  and  given  stresses  there  is 
a  definite  ratio  of  length  to  depth  for  equal  strength  as  given 


§  106.] 


WEB  REINFORCEMENT. 


133 


in  Chap.  Ill,  Art.  91,  but  by  reason  of  the  variation  in  shearing 
strength  due  to  the  direct  effect  of  moment  and  amount  of 
steel,  these  formulas  can  be  considered  as  only  roughly  approxi- 
mate. 

106.  Methods  of  Web  Reinforcement. — There  are  in  use 
many  methods  of  placing  steel  in  the  web  so  as  to  reinforce  it 
against  inclined  tension  failure.  The  various  methods  may,  for 
convenience,  be  divided  into  three  groups:  (1)  Reinforcing 
metal  placed  at  an  inclination;  (2)  Reinforcing  metal  placed 
vertically;  (3)  Miscellaneous  methods. 


Moment 


Shear 

Fia.  49. 

(1)  Theoretically  the  most  effective  way  to  reinforce  against 
tension  failure  in  any  direction  is  to  place  reinforcement  across 
the  lines  of  rupture,  or  in  the  direction  of  the  maximum  tensile 
stresses.  In  the  case  of  web  tension  the  lines  of  maximum 
stress  vary  in  direction,  but  it  is  not  practicable  or  necessary 
to  have  the  inclination  of  the  reinforcing  rods  exactly  the  same 
as  the  lines  of  maximum  tension,  and  various  arrangements 
will  serve  to  accomplish  the  purpose.  The  most  common 
method  is  to  use  several  rods  for  the  horizontal  reinforcement 
and  then  to  bend  a  part  of  these  upwards  as  they  approach  the 
end,  where  they  are  not  needed  to  resist  bending  stresses.  Such 
an  arrangement  is  shown  in  Fig.  50,  (a)  and  (&).  Separate 


134  TESTS  OF  BEAMS  AND  COLUMNS.  [Cn.  IV. 

inclined  rods  may  also  be  used,  attached  or  not  to  the  horizontal 
bars.  The  " stirrups"  commonly  placed  in  a  vertical  position 
may  thus  be  inclined.  Special  forms  of  bars  may  be  used,  as 
the  Kahn  bar,  Fig.  7,  p.  31,  in  which  strips  are  sheared  from 
the  main  bar  and  bent  up. 

(2)  Vertical  reinforcement  has  long  been  the  established 
practice  in  European  work  where  the  experience  has  extended 
over  many  years.  It  has  proven  its  effectiveness  and  in  con- 
nection with  bent  rods  has  many  practical  advantages.  Vertical 
reinforcement  usually  consists  of  some  form  of  bent  rod  or 
band  styled  a  " stirrup",  placed  as  shown  in  Fig.  50,  (c)  and 
(d).  The  Hennebique  system,  widely  and  successfully  used, 
employs  both  the  inclined  rods  and  the  vertical  stirrup  (see 
Fig.  85,  Art.  162).  Combined  with  bent  rods  many  arrange- 
ments of  stirrups  are  possible,  especially  in  continuous-girder 
constructions,  the  chief  object  being  to  secure  good  connection 
of  stirrup  to  top  and  bottom  steel. 

(3)  Some  form  of  web  of  woven  wire  or  expanded  metal 
may  be  used  for  web  reinforcement,  and  still  other  arrange- 
ments of  wire  or  rods  employed  as  illustrated  in  Fig.  50,  (e), 
(/),  and  (g).  In  (</),  representing  the  Visintini  system,  the  beam 
is  made  in  to  a  truss  in  which  the  chords  and  the  tension  diagonals 
are  reinforced. 

107.  Action  of  Web  Reinforcement. — To  aid  in  appreciating 
the  action  of  steel  placed  in  various  ways,  consider  the  typical 
diagonal  tension  failure,  Fig.  51,  as  it  occurs  where  only  hori- 
zontal rods  are  used.  The  inclined  crack  at  a  usually  appears 
first,  due  to  rupture  of  the  concrete  in  tension.  To  assist  in 
preventing  this  rupture  in  its  initial  stage  the  most  efficient 
reinforcement  would  be  such  as  supplied  by  the  inclined  rod  1, 
fastened  to  or  looped  about  the  horizontal  bar,  or  by  the  bent 
end  of  one  of  the  horizontal  bars.  Reinforcement  in  this  direc- 
tion is  in  a  position  to  take  stress  immediately.  The  vertical 
rod  2  can  hardly  be  as  effective  as  the  inclined  rod  in  preventing 
initial  rupture,  for  so  long  as  the  concrete  is  intact  the  deforma- 
tion on  a  vertical  line  is  practically  zero,  owing  to  the  combined 


107.] 


WEB   REINFORCEMENT. 


135 


(a) 


(6) 


IU 


(c) 


(d) 


(e) 


<  (£0 

FIG.  50.— Some  Methods  of  Web  Reinforcement. 


136  TESTS  OF  BEAMS  AND  COLUMNS.  [Cn.  IV, 

action  of  web  tension  and  web  compression  at  right  angles  to 
each  other.  Unless  the  unit  stresses  in  the  steel  be  made  very 
low,  however,  it  is  likely  that  the  concrete  has  received  excessive 
tensile  stress  even  under  working  conditions,  and  may  be  assumed 
to  be  ruptured  more  or  less  in  the  same  manner  as  on  the  tension 
face  of  the  beam  at  points  of  maximum  moment.  At  least  the 
distortion  in  tension  will  be  greater  than  in  compression,  and 
there  will  Ke  a  vertical  movement  of  the  concrete  on  the  left 
of  the  crack,  a,  downwards  with  respect  to  the  part  of  the  right, 
and  the  vertical  rod  2  will  be  brought  into  direct  action  if 
looped  around  or  attached  to  the  horizontal  bars.  Such  a 
rod  may  then  be  more  effective  (allow  of  less  vertical  movement) 
than  the  inclined  rod.  Practically,  there  is  no  great  difference 
in  the  effectiveness  of  the  two  forms  of  reinforcement  if  closely 
spaced  so  as  to  be  in  position  to  prevent  excessive  deformation 
all  along  the  lower  portion  of  the  beam.  To  secure  thoroughly 
effective  reinforcement  in  this  respect  requires  very  careful 
arrangement  of  the  rods  and  faithful  execution  of  the  wor  . 

Vertical  stirrups  spaced  a  distance  apart  equal  to  or 
greater  than  the  depth  of  the  beam  will  give  little  aid  in 
the  prevention  of  diagonal  cracks  between  successive  stirrups 
although  they  may  prevent  final  failure  by  the  extension  of  a 
crack  horizontally  along  the  reinforcing  rods.  Stirrups  should 
be  looped  around  the  horizontal  rods  so  as  to  be  firmly  anchored 
at  their  lower  end  (or  upper  end  at  points  of  negative  moment), 
where  the  stress  is  a  maximum,  but  attachment  to  the  rod  is 
not  necessary,  as  the  office  of  the  stirrup  is  to  prevent  vertical, 
or  nearly  vertical,  distortion.  The  value  of  a  stirrup  unless 
anchored  or  looped  at  the  top  is  limited  by  its  strength  of 
bond,  and  as  its  length  is  not  great  this  point  may  need  con- 
sideration. In  some  tests  at  the  University  of  Wisconsin  final 
failure  has  resulted  from  slipping  of  the  stirrups.  Stirrups 
made  of  small  sections  or  bent  in  loops  are  advantageous  in  this 
respect.  Where  separate  inclined  reinforcement  is  used  there 
is  danger  of  its  slipping  along  the  horizontal  rods  if  the 
inclination  is  too  great. 


§  108.]  WEB  REINFORCEMENT.  137 

Bent  rods  alone  are  apt  to  be  of  limited  value,  owing  to  the 
difficulty  of  providing  rods  close  enough  together.  Conven- 
ience of  horizontal  reinforcement  calls  for  comparatively  few 
rods  of  large  size,  which  provides  too  few  for  effective  diagonal 
reinforcement.  Where  large  rods  are  bent  up  the  length  of 
the  bent  end  should  be  made  sufficient,  by  bending  at  a  small 
angle,  to  develop  the  requisite  bond  strength.  Some  tests  of 
beams  show  failure  of  bond  in  the  case  of  short  bent  rods.  In 


FIG.  51. 

the  case  of  continuous  girders  it  is  convenient  to  extend  the 
bent  rod  horizontally  at  the  top  over  the  support  to  furnish 
tension  reinforcement.  A  very  satisfactory  arrangement  of 
web  reinforcement  is  a  combination  of  bent  rods  and  vertical 
stirrups,  and  especially  is  this  the  case  in  continuous-beam 
construction.  Tests  of  various  arrangements,  so  far  as  the 
authors  have  been  able  to  find,  show  the  best  results  from  this 
method  under  the  ordinary  conditions  and  proportions.  Web 
reinforcement  of  woven  wire  or  expanded  metal  should  give 
good  results. 

1 08.  Effect  of  Stirrups  on  Stress  in  the  Horizontal  rods. — 
A  careful  study  of  the  distribution  of  stress  which  exists 
after  a  beam  begins  to  rupture  on  a  diagonal  line  will  show 
the  fact  that  a  stirrup,  whether  vertical  or  inclined,  will  relieve 
the  stress  in  the  horizontal  rods  at  the  point  of  rupture.  Thus 
in  Fig.  52,  if  the  concrete  no  longer  has  tensile  strength,  the 
value  of  the  tension  T  in  the  horizontal  rods  at  the  line  of 
rupture,  if  unaided  by  the  stirrup  stress  Si  or  $2,  is  equal  to 
Vx/a,  the  same  as  its  value  was  at  section  N  before  rupture 
began.  The  moment  of  the  stress  in  the  stirrup  about  the 
point  A,  whether  the  stirrup  be  vertical  or  inclined,  serves  to 


138 


TESTS  OF  BEAMS  AND  COLUMNS. 


[Cn.  IV. 


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140  TESTS  OF  BEAMS  AND  COLUMNS.  [Cn.  IV. 

reduce  the  value  of  T.    Without  the  stirrup  there  is  therefore 
more  danger  of  failure  of  bond  near  the  ends  of  the  beam. 

109.  Results  of  Tests. — In  Table  No.  8  are  given  in  a  classi- 
fied form  the  most  important  tests  of  rectangular  beams  which 
lend  information  on  web  stresses  and  web  reinforcement.  The 
reference  number  in  the  first  column  refers  to  the  list  of  authori- 
ties on  p.  123.  In  this  table  are  given  the  significant  facts  as 
far  as  practicable,  although  a  detailed  inspection  of  the  reports 
referred  to  is  necessary  for  a  thorough  study  of  the  tests.  The 
kind  of  failure  denoted  as  a  " shear  failure"  is  so  called  for 
convenience;  they  are  diagonal-tension  failures  brought  about 
by  large  shearing  stresses  and  hence  may  be  measured  by  the 
shearing  forces  present.  The  average  shearing  stress  on  a 
vertical  section  at  failure  is  given.  While  the  maximum  shear- 
ing stress  is  somewhat  greater  than  this  (Art.  89)  the  average 
stress  is  practically  as  good  a  standard  of  measure  and  is  much 
more  readily  calculated.  Where  the  failure  was  not  a  shear 
failure  the  figures  for  shearing  stress  are  valuable  as  indicating 
what  the  maximum  stresses  were,  although  the  beam  may 
have  withstood  still  larger  stresses  if  failure  had  not  occurred 
in  some  other  way. 

Straight  Reinforcement  Only. — The  tests  of  Professor  Talbot, 
Professor  Marburg,  and  Mr.  Harding  give  values  of  from  95 
to  128  lbs/in2  under  quite  a  variety  of  conditions.  Mr.  Carson, 
with  specially  good  concrete,  secured  values  of  about  200  in  the 
case  of  high-elastic-limit  deformed  bars  and  182  for  plain  bars, 
which,  however,  failed  in  tension.  In  the  University  of  Wiscon- 
sin tests  on  overhanging  beams,  which  represented  beams  of 
great  depth,  those  with  straight  bars  gave  a  value  of  161  lbs/in2 
and  double  reinforced  beams  values  from  155  to  194  lbs/in2, 
depending  upon  the  per  cent  of  steel  used. 

As  stated  in  Art.  102  the  amount  of  horizontal  steel  has  a 
direct  bearing  on  shear  failures  for  the  reason  that  large  areas 
of  steel  with  low  unit  stresses  permit  less  extension  of  the  con- 
crete than  small  areas  with  high  working  stresses.  This  effect 
is  shown  in  a  marked  manner  in  a  series  of  tests  made  at  the 


§  109.J  WEB  REINFORCEMENT,  141 

University  of  Wisconsin  on  small  mortar  beams  of  1  •  3  mixture. 
The  beams  were  3"X4J"  in  cross-section  and  4  ft.  span  length. 
Loads  were  applied  at  two  points  a  varying  distance  apart. 
Only  straight  reinforcement  was  used,  amounting  to  1.41%. 
The  tensile  strength  of  the  material  was  high,  being  490  lbs/in2. 
The  results  were  as  follows : 

Distance  Apart  Average  Shearing 

of  Loads.  Stress. 

Centre  Load.  Lbs/in2. 

177 

12"  200 

24"  220 

32''  316 

36"  512 

40"  850 

44"  1035 

The  increase  in  strength  as  the  loads  approached  the  supports 
must  be  due  largely  to  the  decrease  in  moment  stress  and 
consequent  distortion,  which  is  essentially  what  occurs  when 
large  areas  of  steel  and  low  working  stresses  are  used. 

Beams  with  Web  Reinforcement. — Mr.  Harding's  tests  included 
only  bent  rods,  and  with  these  very  considerably  higher  ulti- 
mate values  were  obtained  than  for  straight  rods,  averaging  for 
the  three  groups  190  lbs/in2.  Plain  bars,  bent,  gave  tension 
failures,  these  bars  being  of  lower  elastic  limit  than  the  deformed 
bars.  These  results  are  therefore  of  negative  value.  In  some 
of  Mr.  Harding's  tests  the  inclined  bars  pulled  out,  the  bent 
ends  being  relatively  short,  as  indicated  in  the  sketches.  An 
inspection  of  the  deflection  curves  of  these  beams  will  show 
that  those  in  which  the  rods  were  not  bent  were  the  stiffer 
beams,  owing  to  the  greater  average  amount  of  steel  carrying 
the  bending  moment.  Mr.  Carson's  results  average  227 
lbs/in2  for  curved  bars  and  from  220  to  338  lbs/in2  for 
straight  bars  with  stirrups,  the  strength  increasing  with  in- 
creasing per  cent  of  metal.  The  stirrups  were  1"X|"  straps 
spaced  about  7  in.  apart.  A  reference  to  Table  No.  12  will 


142  TESTS  OF  BEAMS  AND  COLUMNS.  [On.  IV. 

show  the  effect  of  stirrups  on  the  ultimate  strength  and  method 
of  failure  of  beams  reinforced  for  compression.  In  the  tests  of 
Mr.  Withey  the  bent  rods  alone  gave  258  lbs/in2  and  stirrups 
alone  averaged  240,  while  the  combination  gave  334,  with  a 
tension  failure,  showing  still  greater  web  stresses  possible. 
Expanded  metal,  as  used,  proved  too  weak,  as  it  pulled  apart 
at  a  shearing  value  of  240  lbs/in2.  T-beam  tests  described  in 
Art.  110  indicate  that  a  value  of  300  lbs/in2  may  readily  be 
reached  with  stirrups  and  bent  rods  even  with  a  relatively  poor 
concrete. 

The  importance  of  tensile  strength  in  the  concrete  should 
be  noted  in  this  connection,  as  the  diagonal  tension  or  shear 
failure  is  the  one  to  be  most  feared  and  therefore  most  care- 
fully guarded  against. 

no.  Tests  on  T-Beams. — The  reinforcing  of  T-beams  re- 
quires special  care  in  providing  against  shearing  stresses.  Where 
a  floor  slab  forms  the  upper  part  of  a  beam  there  will  usually 
be  ample  strength  in  compression  for  any  depth  likely  to  be 
selected.  The  design  of  the  stem  of  the  T,  or  the  beam  below 
the  slab,  is  therefore  largely  a  question  of  providing  sufficient 
concrete  and  reinforcement  to  take  care  of  the  shearing  stresses. 
In  this  case,  therefore,  it  is  important  to  provide  a  strong  web 
for  shearing  stresses,  as  the  strength  in  this  respect  will  com- 
monly determine  its  size.  In  Tables  Nos.  9  and  10  are  given 
the  most  important  tests  on  T-beams  known  to  the  authors. 
The  percentage  of  steel  is  calculated  with  reference  to  a  rectan- 
gular beam  having  a  cross-section  equal  to  the  circumscribing 
rectangle.  The  yield  point  of  the  plain  steel  in  the  tests  of 
Table  No.  9  was  about  37,000  lbs/in2,  and  its  ultimate  strength 
51,000  lbs/in2.  A  load  of  about  19,000  Ibs.  would  stress  the 
steel  in  the  beams  having  .84%  reinforcement  to  the  yield 
point.  This  limit  is  exceeded  only  in  the  last  three  of  the  list. 
In  these  beams  inclined  stirrups  were  used,  placed  in  a  notch 
in  the  bar;  in  all  other  series  the  stirrups  were  placed  vertically. 

Reviewing  these  experiments  we  note,  first,  the  results  with 
straight  bars  and  no  stirrups.  The  beams  having  the  .8-in. 


110.] 


TESTS  ON  T-BEAMS. 


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144  TESTS  OF  BEAMS  AND  COLUMNS.  [Cn.  IV 

rods  and  the  Thacher  bars  developed  a  value  of  119  lbs/in2 
average  shearing  stress,  while  the  .4-in.  rods  developed  198, 
the  difference  being  due  doubtless  to  the  difference  in  bond 
strength,  although  the  previous  experiments  cited  would  indi- 
cate that  not  much  greater  value  than  the  latter  figure  could 
be  expected  from  straight  bars  only. 

Noting  the  next  five  beams,  all  have  straight  rods  and 
vertical  stirrups,  No.  4  having  stirrups  spaced  8"  apart,  while 
the  others  have  a  spacing  of  4"  or  less  near  the  support.  For 
the  former  a  value  of  205  lbs/in2  was  reached,  while  the  three 
others  averaged  341,  all  being  nearly  the  same  despite  the 
variety  of  bars  used.  No.  9  had  bent  bars  and  no  stirrups, 
giving  a  strength  of  252,  while  No.  10  had  bent  rods  and  stirrups 
rather  widely  spaced,  developing  334.  Nos.  11-14  had  inclined 
stirrups  attached  to  the  bars  and  all  but  the  first  gave  high 
values  of  over  450  for  the  shear. 

In  these  tests  it  should  be  noted  that  a  load  of  16,000  Ibs. 
would  develop  in  the  rods  a  theoretical  stress  of  (8000  x  20) /4. 2  = 
38,000  lbs/in2.  For  the  .8-inch  rods  this  would  require  an 
average  bond  strength  of  about  88,000/(2X}X?rX25JX= 
320  lbs/in2,  about  all  that  could  be  expected.  The  .4-inch 
rods  would  be  stressed  one-half  as  much  in  bond.  The  spacing 
of  stirrups  in  No.  10  was  too  great  to  be  entirely  efficient.  The 
inclined  attached  stirrups  gave  the  best  results  in  these  tests, 
but  whether  similar  results  would  be  obtained  where  strength 
of  bond  was  not  in  question  cannot  be  stated.  In  case  of  weak 
bond  an  attached  inclined  stirrup  virtually  adds  much  to  the 
bond  strength  of  the  bar. 

In  Table  No.  10  are  given  further  results  of  tests.  In  the 
first  four  tests  the  supports  were  placed  too  near  the  ends  of 
the  beam  (4  inches)  with  the  result  that  after  the  initial  crack- 
ing the  bars  soon  pulled  out.  After  reducing  the  span  length 
to  5  feet  no  further  trouble  in  this  respect  was  experienced. 
The  results  correspond  closely  with  those  given  in  the  other 
tables. 


§  no.] 


TESTS  ON  T-BEAMS. 


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146 


TESTS  OF  BEAMS  AND  COLUMNS. 


[CH.  IV. 


TABLE  No.  11. 

T-BEAM  TESTS  AT  THE  UNIVERSITY  OF  ILLINOIS* 

Concrete,  1:2:4;  age  about  60  days;  comp.  strength  of  cubes=1820 
lbs/in2. 

Steel:  yield  point  of  plain  round  =17300  Ibs/in2;  of  Johnson  bars= 
36200  lbs/in2. 

Size  of  beams:  thickness  of  flange  =  3^  in.;  thickness  of  web  =  8  in.; 
depth  to  center  of  steel=10  in.;  total  length=ll  ft.;  span  length=10  ft.; 
width  of  flange  varied. 

Stirrups:  made  of  J-in.  Johnson  bars;  five  stirrups  at  each  end  spaced 
6  in.  apart. 

Loads  applied  at  third  points.     All  failures  were  steel  tension  failures. 


Number  and  Size 
of  Rods. 

Average 

Num- 
ber. 

Width  of 
Flange. 
Inches. 

Percent- 
age 
Reinforce- 
ment. 

Total 
Breaking 
Load. 
Pounds. 

Shearing 
Stress  on 
Section 
8"X10" 

Stress  in 
Steel. 
lbs/in«. 

lbs/in2. 

1 

} 

1.05 

3  \ 

"  Johnson 

46700 

293 

64300 

4 

\      16 

1.10 

4  J 

"  Plain  round 

32410 

203 

41500 

7 

J 

1.10 

4  j 

//             U                 11 

30100 

188 

38100 

3 

1 

0.93 

4; 

"  Johnson 

55700 

347 

57500 

6 

1 
}•      24 

0.92 

J5j 
1(2 

"  Plain  round 
bars  bent  up) 

39300 

246 

40700 

8 

J 

0.92 

hi 

1(2 

"  Plain  round 
bars  bent  up) 

40100 

250 

41200 

2 

1 

1.05 

61 

"  Johnson 

80500 

503 

55700 

5 

}•      32 

1.05 

f« 

1(2 

"  Johnson 
bars  bent  up) 

83300 

521 

57400 

9 

1 

0.97 

[7| 

1(3 

"  Plain  round 
bars  bent  up) 

50900 

318 

37600 

Table  No.  11  contains  results  of  recent  tests  by  Professor 
Talbot.  The  maximum  values  of  shearing  stress  are  unusually 
high  and  indicate  very  effective  web  reinforcement.  As  no 
shear  failures  occurred  the  possible  limit  of  strength  of  web 
was  not  determined.  The  very  large  excess  of  stress  in  the 
steel  as  compared  to  the  yield  points  should  be  noted,  due  in 
large  measure  no  doubt  to  the  excessive  compressive  strength 
and  the  thorough  web  reinforcement. 

in.  Conclusions  as  to  Shearing  Strength. — From  the  avail- 
able data  it  would  appear  that  with  ordinary  concrete  and  no 


*  Bulletin  No.  12,  Eng.  Exp.  Station,  Univ.  of  111.,  1907. 


§  112.]  BEAMS  REINFORCED  FOR  COMPRESSION.  147 

web  reinforcement  the  ultimate  average  shearing  strength  is 
about  100  lbs/in2  and  that  this  strength  can  readily  be  in- 
creased by  the  use  of  web  reinforcement  to  300  to  400  lbs/in2. 
The  latter  figure  may,  from  our  present  knowledge,  be  taken  as 
about  the  maximum  value  with  ordinary,  closely  spaced  web 
reinforcement.  It  appears  also  that  the  shearing  strength  of  a 
T-beam  is  about  the  same  as  that  of  a  rectangular  beam  of  the 
same  depth  and  a  width  equal  to  the  width  of  the  stem  of  the 
T.  It  is  to  be  understood  that  the  shearing  stress  is  here  used 
merely  as  a  convenient  measure  of  the  diagonal  tensile  stress, 
which  is  really  the  stress  involved.  This  being  the  case  it  would 
be  incorrect  to  take  any  account  of  the  shearing  strength  of 
the  steel  in  designing  the  reinforcement,  as  is  sometimes  done. 

112.  Beams  Reinforced  for  Compression. —  Generally 
speaking,  it  is  more  economical  to  carry  compressive  stresses 
by  concrete  than  by  steel,  but  limitations  as  to  size  sometimes 
makes  it  desirable  to  strengthen  the  compressive  side  of  a  beam. 
In  cases,  also,  where  both  positive  and  negative  moments  exist 
in  the  same  beam,  either  as  alternating  stresses  or  simultane- 
ously at  different  points,  steel  reinforcement  will  be  used  on 
both  sides  and  its  value  on  the  compressive  side  needs  to  be 
known.  Obviously,  steel  reinforcement  on  the  compression 
side  will  have  little  effect  in  beams  that  would  otherwise  fail 
in  tension  or  shear,  although  there  would  be  some  gain  owing 
to  increased  distance  between  centers  of  tensile  and  compres- 
sive forces.  The  effectiveness  of  steel  in  compression  has  some- 
times been  questioned,  but  the  results  of  tests  on  beams  and 
columns  indicate  that,  in  ordinary  ratios  at  least,  the  steel 
does  its  share  of  work.  Table  No.  12  gives  results  of  tests  on 
double  reinforced  beams  made  at  the  University  of  Wisconsin. 

The  neutral  axis  was  found  by  the  use  of  extensometers, 
after  which  the  stresses  in  steel  and  concrete  at  "load  considered" 
were  found,  assuming  the  compression  in  the  concrete  to  follow 
the  parabolic  law.  Unfortunately,  no  web  reinforcement  was 
used,  so  that  all  the  beams  were  too  weak  in  shear  to  develop 
the  full  compressive  strength,  except  in  the  case  of  the  first 


148 


TESTS  OF  BEAMS  AND  COLUMNS. 


[Cn.  IV. 


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8 


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Si    ^    8 


88888 


CO        CO        CO        CO        CO        CO 


O        "3 


co       CO       CO       CO       CO       CO       CO       CO 
<N<N<M<M<N<M<MC^ 

1O          ^"H  O          ^  O5          CO          C^          CO 

COr^OSGOiOOGOO 
i— ii— ii— (T— ii— iC^i-HCS 

t^*        CO         OS        GO        t^*         O        00         O 

<NC^<M<N<NCOC^CO 

cO        "^f        00        CO        ^O        CO        00        CO 

o^      o^      oo      oo      t^*      r^- 

C50^TP0505^^ 

OOOOOC^-I^H 
<u        o>       Pn  pcj  pn 

OO^'v^^v^ 

<N  rt<  co 

OS 
CSJ        ----- 


punoj 


112.] 


BEAMS  REINFORCED  FOR  COMPRESSION. 


149 


Beam  A 


Bot 
Top 


om  Reinforcement  2.9$ 


Reinforcement  non< 


Beam   B 


Bo 
To 


torn 
»Rei 


Reinforcement 
forcemen 


Eft 


100  \-4J- 


D  fleet  on  in  Inches 
1      02      Ol3      04      Oi5      Ol6      07      08 


.0005  .001  .0015  .002  .0025 

Deformation  per  Unit  Length 

FIG.  53. 


.003 


150 


TESTS  OF  BEAMS  AND  COLUMNS. 


[Cn.  IV. 


two  beams.  The  tensile  stresses  in  the  steel  as  calculated  by 
the  two  methods  agree  very  closely.  The  compression  in  the 
concrete  is  determined  by  subtracting  from  the  total  compres- 
sion the  compressive  stress  in  the  steel.  As  a  check  the  ratio 
of  stress  in  steel  to  stress  in  concrete  has  been  computed.  It 
is  seen  to  be  fairly  constant  and  about  equal  to  the  value  of  n 
for  the  concrete  at  rupture,  as  it  should  be.  These  results 
indicate  that  the  steel  is  taking  its  share  of  stress  and  that  the 
compression  side  of  the  beam  is  strengthened  in  accordance 
with  the  usual  theory.  Obviously,  in  order  to  secure  full  benefit 
of  the  steel  up  to  rupture,  a  fairly  high  elastic  limit  material 
should  be  used. 

Fig.  53  gives  a  typical  set  of  curves  for  the  double  rein- 
forced beams.  Comparing  with  those  shown  in  Art.  100,  it  will 
be  seen  that  these  beams  are  much  stiffer  and  apparently  more 
perfectly  elastic,  as  would  be  expected  from  the  nature  of  the 
reinforcement. 

TABLE  No.  13. 

TESTS  OF  BEAMS  REINFORCED  FOR  COMPRESSION. 

BOSTON  TRANSIT  COMMISSION.* 

Beams  and  material  as  described  in  Table  No.  8.  All  beams  reinforced 
with  \"  corrugated  bars,  with  same  number  top  aid  bottom.  Ltirrups 
1"X|".  spaced  about  1".  Centre  loads. 


Total  Rein- 

Num- 
ber. 

forcement. 

Use  of 

Stir- 
rups. 

Load  at 
First 
Sign  of 
Failure. 
Lbs/in2. 

Ulti- 
mate 
Load. 
Lbs/m2. 

M 

bd2 

Average 
Shearing 
Stress, 
v'. 
Lbs/in2, 

Kind  of 
Failure. 

Num- 
ber of 

Per- 

Bars. 

cent  age 

72 

4' 

XT_ 

9920 

10980 

513 

126 

Tension 

78 

4 

1/JO 

No 

11424 

14148 

660 

162 

'  < 

71 

4 

.62 

Voa 

11000 

16506 

766 

188 

Shear  &  tens. 

77 

4 

i  es 

11224 

15072 

701 

172 

it        «     n 

70 

6^ 

vrn 

14992 

16096 

740 

182 

Shear 

76 

6 

' 

1\0 

16716 

17300 

796 

195 

'  ' 

69 

6 

2.44  < 

•\r 

17724 

23972 

1106 

272 

Tension 

75 

6, 

Yes 

14476 

21284 

990 

244 

t  ( 

68 

8^ 

1 

fciY 

19044 

19044 

880 

215 

Shear 

74 

8 

3OC    J 

JNo 

17200 

18584 

854 

210 

(  i 

67 

8 

.25  < 

Voa 

21200 

30168 

1400 

344 

Tension 

73 

8. 

/ 

i  es 

22132 

29178 

1347 

332 

«  i 

*  Tenth  Annual  Report,  1904. 


§112, 


TESTS  OF  COLUMNS. 


151 


Table  No.  13  gives  results  of  tests  on  double-reinforced  beams 
by  the  Boston  Transit  Commission.  The  table  is  of  value 
mainly  in  showing  the  benefit  of  stirrups.  Crushing  failures 
were  obtained  in  but  few  cases  in  this  series  of  tests,  even  where 
no  compressive  reinforcement  was  used,  so  that  little  advantage 
could  be  expected.  It  should  be  noted  that  where  stirrups  are 
not  used  the  results  shown  in  this  table  are  very  nearly  the 
same  as  those  of  Table  No.  12,  although  the  quality  of  the 
concrete  in  the  latter  case  was  much  inferior.  Conditions  were 
such  that  the  full  strength  of  the  concrete  was  not  developed  in 
the  tests  of  Table  No.  13. 


TABLE  No.  14. 

TESTS  OF  PLAIN  CONCRETE  COLUMNS. 
WATER-TOWN  ARSENAL,  1903-1905. 

All  columns  were  8  ft.  high  and  ranged  from  10  in.  in  diameter  to  12  in. 
square.     The  age  of  the  concrete  ranged  from  5  to  8  months. 


Kind  of  Concrete. 

Crushing  Strength,  Lbs/in2. 

Results  of  Indi- 
vidual Tests.* 

Average 
Crushing 
Strength. 

1  '  1  mortar 

/5011-f          \ 
{  4320              / 
3652     2488 
2062     2692 
/  1564     1471  1 
\  1050              / 
1038     1082 
1525     1720 
3900 
1506     1710 
/  1750     1990  1 
1  1413              / 
462       700  1 
1260              | 
1350              1 
750     1446  / 
871 
/  1060              \ 
1    698              / 

4665 

3070 
2377 

1362 

1060 
1622 
3900 
1608 

1718 
807 

1182 

871 
879 

1:2       "                  .      . 

1:3       "      

1:4       "       

1-5       " 

1-1-2  (pebbles)     . 

1:1:2  (trap  rock)  

1:2:4  (pebbles)  

1-2-4  (trap-rock) 

1:3:6  (pebbles)  
1:3:6  (trap-rock)  

1:2:4  (cinders)  

1*3  -6  (cinders) 

*  Where  two  lines  of  values  are  given,  those  in  the  first  line  are  results  obtained  in 
the  1904  series,  those  in  the  second  line  are  from  the  1905  series. 


152 


TESTS  OF  BEAMS  AND  COLUMNS. 


[On. 


COLUMNS. 

113.  Tests  of  Plain  Concrete  Columns. — The  best  series  of 
tests  which  have  been  made  on  columns,  to  the  authors'  knowl- 
edge, are  those  made  at  the  Watertown  Arsenal,  and  reported 
in  Tests  of  Metals,  1904,  and  subsequent  volumes.  The  prin- 
cipal results  on  plain  concrete  are  given  in  Table  No.  14. 

TABLE  No.  15. 

TESTS  OF  REINFORCED  COLUMNS. 

MASSACHUSETTS  INSTITUTE  OF  TECHNOLOGY. 


Num- 
ber. 

Cross- 
section. 

Ratio: 
Length 
Diam. 

Number 
of  Rods 
and  Size 
(Square). 

Plain 
or 
Twisted. 

Area  of 

Steel, 
Sq.  In. 

Percent- 
age of 
Rein- 
force- 
ment. 

Crushing 
Strength. 
Lbs/in2. 

1 

8"X8" 

25.5 

1    I" 

P 

1 

1.56 

1670 

2 

25.5 

1    1" 

T 

1.56 

1985 

3 

18.0 

1  V 

P 

1.56 

1560 

4 

18.0 

1  ]" 

T 

1.56 

1970 

5 

9.0 

1  1" 

P 

1.56 

2160 

6 

9.0 

1  V 

T 

1.56 

2080 

7 

25.5 

i  ii" 

P 

.56 

2.44 

2125 

8 

25.5 

i  il" 

T 

1.56 

2.44 

2410 

9 

25.5 

4  f" 

P 

2.25 

3.51 

2840 

10 

25.5 

4  I" 

T 

2.25 

3.51 

2610 

11 

18.0 

4  f" 

T 

2.25 

3.51 

2300 

12 

18.0 

4  f" 

P 

2.25 

3.51 

2390 

13 

9.0 

4  1" 

T 

4.0 

6.25 

2470 

14 

9.0 

4  1" 

P 

4.0 

6.25 

3810 

15 

10"X10" 

20.4 

1   1" 

P 

1 

1 

2150 

16 

•7.2 

1   1" 

P 

1 

1 

2000 

17 

7.2 

1   1" 

T 

1 

1 

2284 

18 

14.4 

1   1J" 

T 

1.56 

1.56 

2620 

19 

14.4 

1   1J" 

P 

1.56 

1.56 

2570 

20 

14.4 

4  |" 

T 

2.25 

2.25 

3000 

21 

14.4 

4  f" 

P 

2.25 

2.25 

2740 

These  tests  indicate  an  average  strength  for  1^2:4  concrete 
of  1600  to  1700  lbs/in2,  with  no  excessive  variation  in  indi- 
vidual tests.  For  the  weaker  mixture,  1;3:6,  the  individual 
tests  are  much  more  at  variance,  indicating  greater  unrelia- 
bility. The  great  strength  of  very  rich  mortars  is  noteworthy, 
and  this  fact  is  borne  out  by  experiments  on  columns  slightly 
reinforced.  Considering  relative  cost,  a  rich  mortar  may  often 
be"  the  more  advantageous.  Thus,  for  example,  if  cement,, 


§114.] 


TESTS  OF  COLUMNS. 


153 


sand,  and  stone  cost  respectively  $2.00,  $0.75  and  $1.00  per 
unit,  and  the  cost  of  mixing  and  placing  be  $1.50,  the  net  cost 
of  a  cubic  yard  of  1:2:4  concrete  will  be  about  $5.85,  and  the 
cost  of  a  yard  of  1:1  mortar  will  be  about  $12.00,  or  slightly 
more  than  double,  while  the  strength  is  about  three  times  as 
great.  Similarly,  the  cost  of  a  1 :2  mortar  is  about  $8.85,  while 
it  has  nearly  double  the  strength  of  a  1:2.4  concrete. 

114.  Tests  on  Columns  with  Longitudinal  Reinforce- 
ment.— The  results  of  a  valuable  series  of  experiments  made 
at  the  Massachusetts  Institute  of  Technology  are  given  in 
Table  No.  15.*  The  concrete  was  1:3:6  broken  stone  concrete; 
the  rods  were  partly  plain  square  rods  and  partly  twisted 
rods,  the  strength  of  the  plain  rods  being  56,000-60,000 
lbs/in2,  and  of  the  twisted  rods  about  80,000  lbs/in2.  Where 
single  rods  were  used  they  were  placed  in  the  centre,  and  where 
four  rods  were  used  they  were  placed  in  the  form  of  a  square 
one-half  the  dimensions  of  the  column.  The  columns  were 
approximately  thirty  days  old. 

Grouping  these  tests  in  accordance  with  the  amount  of 
reinforcement  we  have  the  following  average  values : 


Calculated 

Per  Cent 

Average  Strength, 

Strength, 

Reinforcement. 

Lbs/in2. 

/=  1470(1+  19p). 

Lbs/in2. 

{ 

1.56 

1904 

1904 

8"X8"  columns. 

2.44 

2267 

2170 

Average  length  =  12.4  ft.  1 

3.51 

2535 

2450 

I 

6.25 

3140 

3250 

/=  1800(1  +I9p). 

10"X10"  columns.             { 
Average  length  =11.0  ft.  j 

1.0 
1.56 
2.25 

2145 
2452 
2870 

2145 
2320 
2600 

It  is  evident  that  the  larger  columns  are,  for  like  reinforce- 
ment, stronger  than  the  smaller  columns,  showing  an  effect 
either  of  ratio  of  length  to  diameter  or  of  diameter  directly. 
Little  difference  is  observed  between  plain  and  twisted  bars. 
The  effect  of  amount  of  reinforcement  can  be  observed  by  con- 

*  Trans.  Am.  Soc  C.  E.,  Vol.  L,  1903,  p.  487. 


154 


TESTS  OF  BEAMS  AND  COLUMNS. 


[Cn.  IV. 

sidering  each  size  separately.    The  results  have  been  studied 
on  the  basis  of  the  theoretical  formula  of  Art.  95,  Chapter  III, 

P' 

.     •     (1) 


in  which  P'/P  represents  the  ratio  of  the  strength  of  the  rein- 
forced to  that  of  the  plain  concrete  column. 

No  results  are  given  for  plain  concrete  columns,  but  as- 
suming that  the  column  with  the  lowest  percentage  of  steel 
follows  the  theoretical  law  the  strength  of  the  ideal  plain  con- 
crete column  is  calculated  to  be  1470  lbs/in2  for  the  first 
group  and  1800  lbs/in2  for  the  second  group,  making  n  =  2Q. 
Taking  these  values  then  as  a  basis  the  results  are  plotted  in 
Fig.  54.  Abscissas  represent  per  cent  of  reinforcement  and 


B£U 

o>24P 
1 

SO 

-X 

^* 

s^ 

-..0 
c 

.X 

^. 

oj^ 

X**^ 

of  Strength  to  Pla 

L_JL_iL_! 

> 

's^ 

.s' 

•^ 

,  ^ 

<^^ 

^ 

• 

^ 

^ 

o  i-t" 

*.-. 

0^ 

.^ 

•  8" 

a  8  "Col 

umns 

1.00 

^ 

OlO 

KlQ" 

^ 

•z. 

3  A  o 

Percentage  of  Beinfoxcemejit 


FIG.  54.— Tests  of  Reinforced  Columns.     (Mass.  Inst.  of  Technology.) 

ordinates  the  relative  strengths,  that  of  the  ideal  plain  con- 
crete being  100.  The  theoretical  relation  is  shown  by  the 
straight  line  drawn  for  n=20.  This  value  of  n  corresponds  to 
a  value  of  Ec  of  1,500,000,  which  would  be  a  reasonable  value 
at  rupture  on  the  basis  of  total  deformation,  as  explained  in 
Art.  24.  While  the  results  are  not  sufficiently  numerous  to  be 
at  all  conclusive,  they  do  indicate  that  the  relative  strength 


§114.] 


TESTS  OF  COLUMNS. 


155 


of  such  columns  is  fairly  represented  by  the  theoretical  law. 
Calculated  values  corresponding  to  the  theoretical  lines  of  the 
diagram  are  given  by  the  formulas 


and 


/  =  1470(1  +  19p) 
/  =  1800(l  +  19p), 


These  values  are  given  in  the  table  on  p.  153.  Eliminating  the 
longest  columns  of  the  first  group  a  fairly  correct  value  for  the 
ultimate  strength  of  all  would  be  given  by  /  =  1600(l  +  19p) 
(n  is  assumed  equal  to  20). 

The  following  table  gives  results  of  tests  made  at  the  Water- 
town  Arsenal  on  concrete  columns  reinforced  with  longitudinal 
bars  only.  All  columns  were  8  ft.  long  and  approximately 
12"X12"  square;  age,  3J  to  8  months. 

TABLE  No.  16. 

TESTS  OF  REINFORCED  COLUMNS. 
WATERTOWN  ARSENAL,    1904-1905. 


Reinforcement. 

Com- 

Strength 
of  Plain 

Ratio  of 
Strength 
of  Rein- 

Kind of  Concrete. 

Description. 

Per  Cent. 

pressive 
Strength, 
Lbs/in2. 

Concrete. 
(See  Table 
No.  14.) 

forced 
Concrete 
to  Plain 

Concrete. 

1  *  2  mortar 

8  I"  bars 

2.85 

4200 

3070 

1    37 

1:3       "      

t(        a 

2.87 

3841 

2377 

1.61 

1:4       "      

it        « 

2.86 

3377 

1518 

2.22 

1:5       "      

(t        t< 

2.86 

2813 

1060 

2.65 

1:5       "      

13  f"    " 

4.63  - 

3905 

106G 

3  68 

1:1:2  (pebbles)   .  . 

4  f"  twisted 

1.46 

2890 

1720 

1.68 

1:2:4 

U                    (I 

1.43 

1990] 

1.17 

(              i 

4  t"  Thacher 

1.03 

1990  | 

1.17 

t                      e 

4  f"  corrugated 

.97 

2180  I 

1.28 

t                      t 

4  I"  twisted 

1.45 

1820  }• 

1710 

1.06 

<                      i 

8  J"       " 

2.86 

3160  | 

1.84 

i                      ( 

8  |"  Thacher 

2.09 

2760  1 

1.62 

t                      f 

8  f  "  corrugated 

1.94 

2830  J 

1.66 

1:3:6          '          '.  '. 

4  f  "  twisted 

1.44 

1370 

462 

2.96 

1:3:6  (trap-rock).  . 

8  1"  corrugated 

<  :                     <i 

1.94 
1.93 

2290 
2650 

}  1350 

1.82 

1:2:4  (cinders)  

4  f  "  twisted 

1.45 

2095 

871 

2.40 

1:3:6         "      .... 

4  f  "  bars 

1.42 

1932 

] 

1.82 

<*             i» 

8  f"    " 

2.83 

3100 

1  1060 

j 

2.92 

156 


TESTS  OF  BEAMS  AND  COLUMNS. 


[Cn.  IV. 


On  Fig.  55  are  plotted  the  results  of  the  mortar  tests  and  the 
1:2:4  concrete  in  the  same  manner  as  the  values  in  Fig.  54, 
using  as  a  standard  the  results  on  plain  concrete  given  in  Table 
No.  14.  Average  values  have  been  plotted  for  the  columns 
with  percentages  of  .97  and  1.03  and  of  1.43  and  1.45.  Lines 


* 


3.80 
3.60 
3.40 
3.20 
3.00 
2.80 
2.60 
2.40 

2.00 

/ 

*l-5 

7 

2_ 

/ 

/ 

y* 

7 

y 

/ 

y^ 

/ 

r 

2_ 

/ 

3 

r 

f 

1-5  • 

•v/ 

/ 

^x 

/ 

/^ 

x 

y 

ij22 

* 

x1^ 

"7 

y 

^ 

/ 

'  ,J 

JZ 

<^ 

/ 

> 

7 

2_ 

,x 

^^ 

x^ 

/ 

y 

>x^ 

<$& 

s^ 

1  80 

/ 

"2_ 

cy 

^, 

s^ 

y 

'    / 

X^ 

^^ 

•^^ 

1  60 

/  ( 

^x 

1 

S^^x- 

/  / 

X 

^" 

1  40 

/ 

f  > 

^^ 

•  M 

otarT 

esta 

// 

/ 

^ 

•  1 

-2 

01- 

J.-4-Cor 

crete 

1  20 

y 

XX 

^^ 

/£ 

^^ 

r 

1.00 

e^ 

012345 

6 

Percentage  of  Reinforcement 
FIG.  55. — Tests  of  Reinforced  Columns.     (Watertown  Arsenal.) 

have  also  been  drawn  representing  the  theoretical  relations  for 
different  values  of  n.  In  the  mortar  tests  the  results  show 
that  for  the  poorer  mortars  the  relative  effect  of  the  steel  is 
high,  corresponding  to  what  would  be  obtained  theoretically 
by  using  a  value  of  n  =  40  to  50.  In  the  1:2:4  concretes  the 
results  do  not  vary  widely  from  the  theoretical  results  for 
^  =  30,  or  a  value  of  Ec  at  rupture  of  1,000,000. 


2  114.] 


TESTS  OF  COLUMNS. 


157 


It  is  assumed  in  the  theoretical  discussion  that  the  steel  is 
not  stressed  beyond  its  elastic  limit.  It  is  to  be  noted  that 
in  these  tests  the  stress  on  the  steel  bars  must  have  been  as 
high  as  45,000  to  50,000  lbs/in2,  showing  the  usefulness  of  a 
fairly  high  elastic-limit  steel  in  this  case.  (See  further  dis- 
cussion in  Chapter  V.) 

TABLE  No.  17. 

TESTS  OF  REINFORCED  COLUMNS. 
UNIVERSITY  OF  ILLINOIS,  1906.* 


No. 

Length. 

Cross-section. 

• 

Reinforcement. 

Crushing  Strength. 
Pounds  per  sq.  in. 

Kind. 

Per  cent. 

Individual 
Test. 

Average 
of  Group. 

1 

4  f-in.  rods 

1.20 

1587 

1 

3 

7 

12  ft. 

12"X12" 

(    4  f-in.  rods 
{  12  i-in.  ties 
4  f-in.  rods 

jl.21 
1.21 

1862 
1850 

1809 

11 

j    4  f-in.  rods 
\  12   1-in.  ties 

|  1.21 

1936 

J 

2 

12ft. 

1 

4  f-in.  rods 

1.52 

1577 

1 

6 

« 

4  f-in.  rods 

1.52 

1600 

10 

it 

f    4  f-in.  rods 
\  12  J-in.  ties 

}l.50 

1280 

12 

9ft. 

9"X9" 

j     4  f-in.  rods 
1     9  J-in.  ties 

1.48 

2335 

1710 

14 

12ft. 

4  f-in.  rods 
12  J-in.  ties 

1.50 

1367 

16 

9ft. 

4  f-in.  rods 
9  i-in.  ties 

}  1.49  ' 

1607 

17 

6ft. 

'. 

4  f-in.  rods 

1.47 

2206 

5 

12ft. 

12"X12" 

] 

1710 

8 

it 

9//x9'/ 

2004 

9 
13 

(i 

12"X12'' 

K         tt 

I  Plain 

0 

1610 
1709 

[1550 

15 

6ft. 

1C                    <( 

1189 

18 

9"X9" 

1079 

Table  No.  17  contains  the  results  of  tests  made  by  Profes- 
sor A.  N.  Talbot  at  the  University  of  Illinois.  The  columns 
were  made  of  1:2:3-3/4  concrete  and  plain  steel  of  39,800 
pounds  per  square  inch  elastic  limit.  The  age  was  from  59  to 
71  days.  Comparing  the  reinforced  with  the  plain  concrete, 


*  Bulletin  No.  10,  Engineering  Exp.  Sta.,  1907. 


158  TESTS  OF  BEAMS  AND  COLUMNS.  [Cn.  IV- 

the  average  strength  of  the  12"xl2"  columns  with  1.2  per 
cent  reinforcement  is  about  1.17  times  as  great,  and  the  9"X9" 
columns  with  1.5  per  cent  reinforcement  is  about  1.10  times 
as  great.  These  tests  indicate  a  less  effect  of  reinforcement 
than  some  of  the  other  tests  quoted.  The  smaller  cross-section 
of  the  columns  containing  the  larger  amount  of  reinforcement 
may  have  been  the  cause  of  the  lower  strength  of  this  group. 
It  is  important  to  note  the  wide  variation  in  the  individual 
results  of  these  and  other  tests;  they  indicate  what  may  be 
expected  in  practice,  and  show  clearly  the  necessity  of  adopt- 
ing conservative  values  of  working  stress.  Careful  measurement 
of  distortions  showed  that  the  ratio  of  stress  in  steel  to  stress 
in  concrete  varied  from  about  14  at  the  ^beginning  to  about  27 
at  rupture,  taking  average  values.  The  low  value  for  ultimate 
strength  of  the  reinforced  columns  appeared  to  be  due  to  a 
lower  actual  crushing  strength  of  the  concrete  in  these  columns 
than  in  the  plain  columns. 

115.  Effect  of  Length  of  Column  on  Compressive 
Strength. — Comparing  the  results  on  plain  concrete  columns, 
p.  151,  with  the  tests  on  cubes,  pp.  11-14,  it  is  evident  that 
the  strength  of  the  column  is  materially  less.  While  there  is 
thus  a  very  considerable  reduction  of  strength  as  compared  to 
the  cube,  there  appears  to  be  little  difference  in  the  strength  of 
columns  of  various  lengths  up  to  15  to  20  diameters.  A  series, 
of  tests  made  at  the  Watertown  Arsenal  *  for  the  Aberthaw 
Construction  Co.  on  12"  Xl2"  columns  gave  practically  the 
same  results  for  all  lengths  from  2  ft.  to  14  ft.,  the  average  of 
all  being  957  lbs/in2  for  hand-mixed  and  1099  lbs/in2  for 
machine-mixed  concrete.  The  temperatures  were,  however, 
low,  and  the  results  are  riot  a  fair  criterion  as  to  absolute 
strength. 

In  the  tests  of  Table  No.  15  the  difference  in  average  results 
upon  the  8"X8"  columns  and  those  on  the  10"X10"  size  is 
marked.  But  comparing  results  for  each  size  among  them- 

*  Tests  of  Metals,  1897. 


§  116.]  HOOPED  COLUMNS.  159 

selves  there  is  little  or  no  effect  noticeable  up  to  25  diameters. 
Numbers  2  and  3  are  reported  as  having  failed  by  buckling, 
but  these  average  practically  the  same  as  Nos.  1  and  4.  From 
these  tests  it  would  appear  therefore  that  no  account  need  be 
taken  of  length  of  column  below  about  25  diameters,  although 
caution  should  be  used  in  accepting  these  results  as  conclu- 
sive. Twenty  diameters  would  seem  to  be  a  safe  length  below 
which  a  uniform  working  stress  may  be  used.  The  working 
strength  should,  however,  be  taken  materially  below  that  for 
beams.  Greater  lengths  than  20  diameters  are  rarely  needed 
for  important  members.  Where  necessary  they  maybe  designed 
by  the  usual  column  formulas,  the  reinforcement  being  in 
this  case  of  great  importance. 

116.  Hooped  Concrete  Columns. — If  a  compression  mem- 
ber be  reinforced  by  banding  or  winding,  such  a  reinforcement 
will  raise  the  ultimate  strength  by  preventing  lateral  expan- 
sions under  the  compressive  forces.  It  was  shown  in  Art.  96 
that  under  this  system  of  reinforcement  the  steel  cannot  be 
stressed  as  high  under  low  loads  as  in  the  case  of  longitudinal 
reinforcement,  and  that  while  distortion  may  be  great  the 
ultimate  strength  may  be  high.  The  strengthening  effect  of 
banding  will  then  depend  upon  the  amount  of  metal  used  and 
its  resistance  to  expansion  as  measured  by  the  relative  rigidity 
of  steel  and  concrete.  As  in  the  case  of  longitudinal  reinforce- 
ment the  greatest  relative  effect  occurs  with  poor  concrete  of 
low  m.odulus.  Tests  on  columns  show  that  in  general  such 
columns  deform  or  compress  more  than  those  with  longitudinal 
reinforcement. 

In  1902  and  1903  Considere  *  published  certain  tests  made 
on  columns  reinforced  by  spirally  wound  wire  and  by  longi- 
tudinal rods  or  wire.  His  most  important  results  were  those 
obtained  upon  a  number  of  octagonal  columns  5.9  in.  short 
diameter.  As  a  result  of  these  and  other  tests,  as  well  as  from 
a  theoretical  basis,  he  came  to  the  conclusion  that  steel  in  the 

*  Genie  Civil,  1902. 


160 


TESTS  OF  BEAMS  AND  COLUMNS. 


[Cn.  IV. 


form  of  spiral  reinforcement  was  2.4  times  as  efficient  as  in  the 
form  of  longitudinal  reinforcement,  presuming  the  spacing  of 
the  wire  to  be  not  great  (}  to  -fa  of  the  diameter  of  the  spiral) 
and  that  ordinary  mild  steel  be  used.  It  was  found  also  de- 
sirable to  use  a  small  amount  of  steel  in  the  form  of  longitudinal 
reinforcement.  Tests  on  the  elastic  properties  showed  con- 
siderable deformation  and  set,  but  after  the  first  application  of 
a  load  the  column  is  relatively  rigid,  with  greatly  increased 
value  of  E. 

Table  No.  18  gives  results  of  an  important  series  of  experi- 
ments on  hooped  columns  conducted  by  Bach.*     The  columns 


•2,60 


2'40 


c  2.20 

a 

•32.00 


of  Stre 


Percentage  of  Reinforcement 

FIG    56. — Tests  of  Hooped  Columns.     (Bach) 

were  of  octagonal  form  with  short  diameter  equal  to  275  mm. 
and  height  of  1  m.  The  concrete  was  1 :  4  gravel  concrete  5-6 
months  old.  Each  result  is  the  average  of  three  tests,  except 
in  the  case  of  the  unreinforced  concrete,  where  four  tests  were 
made.  The  steel  was  mild  steel.  The  results  in  the  last  column 
do  not  indicate  as  great  increase  in  strength  as  might  be  expected 
and  the  relative  effect  of  longitudinal  and  spiral  reinforcement 
does  not  appear  to  be  greatly  different. 


*  Quoted  from  Morsch,  Eisenbetonbau,  p.  70. 


§  116. 


HOOPED  COLUMNS. 


161 


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162  TESTS  OF  BEAMS  AND  COLUMNS.  [Cn.  IV. 

To  exhibit  these  results  in  a  graphical  manner  they  have 
been  plotted  in  Fig.  56  in  the  same  manner  as  those  in  Figs.  54 
and  55.  The  strength  of  1890  lbs/in2  for  the  plain  concrete 
column  has  been  taken  as  unity.  The  total  percentages  of 
reinforcement  have  been  taken  as  abscissae.  All  the  results  of 
each  group  have  been  connected  by  a  dotted  line  to  aid  in 
studying  them.  Theoretical  lines  for  longitudinal  reinforce- 
ment have  been  drawn  for  n  =  2Q  and  25.  A  careful  study  of 
these  tests  in  comparison  with  those  of  Figs.  54  and  55  fails  to 
show  any  considerable  difference  in  relative  strength  for  a  given 
total  amount  of  reinforcement.  The  results  appear  to  be  about 
the  same.  They  do  not,  therefore,  show  any  considerable  superi- 
ority of  spiral  over  longitudinal  reinforcement.  Group  III,  for 
example,  has  relatively  more  spiral  reinforcement  than  group 
II,  but  its  curve  is  lower.  Groups  V  and  VI  are  both  rela- 
tively weak,  probably  owing  to  the  wide  spacing  of  the  spirals. 

Tests  made  at  the  Watertown  Arsenal  in  1905  and  reported 
by  Mr.  James  E.  Howard  *  showed  a  large  effect  from  hoops 
consisting  of  riveted  bands  1.5"X.12".  Results  on  1:2:4 
columns  10J  in.  diameter  X  8  ft.  long,  5-6  months  old,  were  as 
follows : 

Strength, 
lbs/ina. 

Plain  concrete  columns 1413 

13  hoops 2232 

13  hoops,  4  angle-bars. 3029 

25  hoops 3428 

25  hoops,  4  angle-bars 4189 

47  hoops 5289 

The  size  of  the  angles  was  not  stated.  The  amount  of  steel  in 
the  hoops  is  approximately  1%  for  13  hoops,  1.8%  for  25  hoops, 
and  3.4%  for  47  hoops.  Compared  to  the  plain  concrete 
column  the  strengthening  effect  of  the  hoops  is  relatively  greater 
than  the  longitudinal  reinforcement  previously  discussed. 

*  Proc.  Am.  Soc.  Test  Materials,  Dec.,  1906. 


f  116.].  HOOPED  COLUMNS.  163 

Important  tests  by  Professor  A.  N.  Talbot  on  hooped  col- 
umns *  showed  greatly  increased  ultimate  strength  but  little  or 
no  effect  of  the  reinforcement  for  loads  up  to  the  usual  ultimate 
strength  of  plain  concrete.  The  general  results  were  in  accord- 
ance with  the  discussion  of  Art.  96.  The  total  deformation  at 
failure  was  very  great,  amounting  to  eight  or  ten  times  that 
for  plain  concrete;  the  lateral  deflection  was  also  large.  Crack- 
ing or  scaling  of  the  thin  exterior  shell  of  concrete  began 
at  a  load  about  equal  to  that  causing  failure  in  the  plain 
concrete. 

As  regards  ultimate  strength,  the  effect  of  the  reinforcement 
was  from  two  to  four  times  as  great  as  would  be  caused  by 
the  same  amount  of  longitudinal  reinforcement.  The  following 
formulas  were  found  to  express  approximately  the  ultimate 
strength  in  terms  of  percentage  of  steel  used* 

for  mild  steel,     P'  =  1600  +  65,000  p; 
for  high  steel,     P'  =1600+ 100,000  p; 

where  P'  =  strength  per  square  inch,  and  p  =  percentage  of  steel 
with  reference  to  ,the  concrete  core  within  the  hoops.  The 
strength  of  plain  concrete  is  assumed  to  be  1600  pounds  per 
square  inch. 

The  tests  given  herein,  excepting  possibly  those  of  Bach, 
show  that  the  effect  of  hooping  is  to  render  the  column 
"  tough"  and  to  increase  greatly  its  ultimate  resistance.  The 
accompanying  deformations  are,  however,  large,  and  there 
appears  to  be  little  or  no  aid  rendered  until  a  load  is  reached 
about  equal  to  the  ultimate  strength  of  plain  concrete.  This 
makes  it  difficult  to  combine  effectively  longitudinal  and  hoop 
reinforcements.  This  question,  together  with  the  subject  of 
working  stresses,  is  further  discussed  in  Chapter  V. 


*Proc.  Am.  Soc.  Test.  Materials.  1907:  Eng.  Record,  Aug.  10, 1907,  p. 
145. 


164 


TESTS  OF  BEAMS  AND  COLUMNS. 


[Cn.  IV, 


117.  Fatigue  Tests  of  Reinforced  Concrete. — Important 
experiments  conducted  by  Professor  J.  L.  Van  Qrnum*  on 
reinforced  beams  indicate  an  effect  under  repeated  application 
of  loads  similar  to  that  which  he  found  for  mortar  and  concrete 
in  compression  as  mentioned  on  p.  25.  In  the  case  of  beams 
the  failure  under  repeated  loads  appeared  to  be  largely  a  gradual 
fracture  in  diagonal  tension,  ending  with  a  compression  failure. 
The  number  of  repetitions  required  to  produce  failure  varied 


Percentage  of  Repetition  Load,  in  terms  of  Maximum  Strength  ^ 

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FIG.  57. 


as  ooo 


with  the  load  applied,  rupture  being  ultimately  produced  after 
several  thousand  repetitions  for  loads  as  low  as  55  and  60%  of 
the  usual  ultimate  strength.  The  most  important  of  his  results 
are  indicated  in  Fig.  57,  taken  from  his  paper,  showing  the 
number  of  repetitions  required  to  produce  failure  at  various 
values  of  maximum  load  in  percentage  of  the  usual  ultimate 
load. 

The  change  in  the  modulus  of  elasticity  was  also  investigated, 
and  it  was  found  that  under  repeated  loads  not  ultimately 


*  Trans.  Am.  Soc.  C.  E.,  1907,  LVIII,  p.  294. 


§  117.]  HOOPED   COLUMNS,  165 

causing  rupture  the  concrete  soon  became  perfectly  elastic,  with 
a  value  of  the  modulus  of  about  two-thirds  of  its  initial  value. 
At  loads  ultimately  causing  rupture  the  modulus  became  for 
a  time  nearly  constant,  but  rapidly  decreased  as  rupture  was 
approached. 

These  tests  indicate  that  concrete  when  repeatedly  loaded 
beyond  about  50%  of  its  ordinary  ultimate  strength  will  not 
remain  indefinitely  elastic  and  will  fail.  This  limit  may  be 
called  the  permanent  elastic  or  fatigue  limit  of  concrete.  It  is 
of  much  importance  in  relation  to  working  stresses. 


CHAPTER  V. 
WORKING  STRESSES  AND  GENERAL  CONSTRUCTIVE   DETAILS. 

ii  8.  Working  Stresses  and  Factors  of  Safety. — In  the  design 
of  steel  structures  it  has  come  to  be  the  practice  to  make  use 
of  definite  working  stresses  rather  than  factors  of  safety.  These 
working  stresses  are  based,  for  the  most  part,  on  the  permanent 
elastic-limit  strength  of  the  material,  although  the  margin  of 
safety  between  the  elastic-limit  and  the  ultimate  strength  (indi- 
cated by  strength  and  ductility)  receives  consideration.  The 
working  stresses  are  made  sufficiently  below  the  elastic  limit  to 
provide  for: 

(a)  Variations  and  imperfections  in  material  and  work- 

manship. 

(b)  Uncalculated   stresses,   such  as   secondary  stresses, 

stresses  due  to  unequal  settlement,  and,  usually, 
those  due  to  temperature  changes. 

(c)  Dynamic  effect  of  live  load  if  not  provided  for  by  an 

allowance  for  impact. 

(d)  Possible  increase  in  live  load  over  that  assumed,  or 

rare  applications  of  excessive  loads. 

(e)  'Deterioration  of  the  structure. 

The  more  accurately  the  various  elements  are  determined  in 
any  case  the  closer  may  the  working  stress  approach  the 
elastic  limit.  Where  the  dynamic  effect  of  the  live  load  does 
not  enter,  or  is  otherwise  fully  provided  for,  and  where  items 
(d)  and  (e)  are  of  small  moment,  working  stresses  for  steel  struc- 
tures will  vary  from  about  one-half  to  two-thirds  the  elastic- 
limit  strength  of  the  material.  Were  it  absolutely  certain  that 

166 


V'  OF  THE  ^ 

UNIVERSITY  J 

OF  / 

^FOBg^ 
§  119.]        WORKING  STRESSES  AND  FACTORS  OF  SAFETY.        167 

the  elastic  limit  of  the  material  would  never  be  exceeded  in 
any  emergency,  then  the  margin  of  strength  between  the  elastic 
limit  and  the  ultimate  strength  would  be  of  no  importance. 
This  is,  however,  not  the  case,  and  under  actual  conditions  of 
service  there  is  a  very  considerable  element  of  safety  in  the 
fact  that  the  ultimate  strength  is  in  most  materials  much 
higher  than  the  elastic  limit.  Stated  in  another  way,  a  designer 
would  never  use  a  working  stress  of  one-half  or  two-thirds  the 
elastic  limit  in  a  material  where  the  ultimate  strength  did  not 
considerably  exceed  this  limit.  While  therefore  the  working 
stresses  are  selected  chiefly  with  reference  to  the  elastic  limit, 
the  ultimate  strength  also  receives  consideration. 

In  recent  years  most  designers  base  their  calculations  on 
certain  working  stresses  selected  as  above  indicated.  Formerly, 
and  to  some  extent  now,  calculations  are  based  on  specified 
"factors  of  safety"  referred  to  ultimate  strengths.  In  either 
case  both  the  elastic  limit  and  the  ultimate  strength  must  be 
considered  in  the  design,  and  experienced  designers  will  arrive 
at  about  the  same  results  by  either  method.  In  reinforced-con- 
crete  design  the  problem  is  complicated  by  the  use  of  two  unlike 
materials  whose  elastic  limits  and  ultimate  strengths  are  not 
similarly  related.  Furthermore,  as  the  materials  are  stressed 
beyond  their  elastic  limits  the  stresses  do  not  necessarily  increase 
in  proportion  to  the  load,  so  that  if  working  stresses  of  one- 
fourth  the  ultimate  are  selected,  for  example,  the  corresponding 
load  may  be  considerably  greater  or  less  than  one-fourth  the 
ultimate  load.  This  condition  makes  it  especially  desirable 
to  consider  ultimate  strength,  and  is  an  argument  for  the  use 
of  the  "  factor-of -safety  "  method. 

119.  Relative  Effect  of  Dead  and  Live  Loads. — The  ten- 
dency of  practice  in  the  treatment  of  live-load  stresses  is  to  re- 
duce them  to  equivalent  dead-load  stresses  by  the  application 
of  some  sort  of  impact  formula  or  by  other  means  of  estimation. 
The  resulting  stresses  are  then  considered  on  the  same  basis 
as  the  usual  dead-load  stresses  and  a  single  set  of  working  stresses 
applied.  This  method  is  simple,  logical,  and  tends  to  facilitate 


168  WORKING  STRESSES.  [On.  V. 

a  proper  adjustment  of  the  design  to  the  conditions.  Separate 
working  stresses  will  give  equally  satisfactory  results  when 
properly  selected,  but  the  system  is  not  as  flexible  or  convenient 
as  the  method  of  the  single  working  stress  with  impact  coeffi- 
cients. 

The  question  of  impact  coefficients,  or  the  relation  between 
live-  and  dead-load  working  stresses,  requires  little  special 
attention  in  connection  with  reinforced  concrete  structures. 
It  is  essentially  the  same  as  it  is  in  the  case  of  steel  structures, 
excepting  as  the  amount  of  impact  may  be  modified  by  the 
structure  itself.  In  steel  railroad  structures  of  short  span,  for 
example,  the  impact,  or  dynamic  effect  of  live  load,  is  usually 
assumed  to  be  about  100%  of  the  live  load  stresses.  Experi- 
ments show  that  this  is  probably  not  too  high  and  that  the 
actual  stresses  from  live  load  may  be  100%  greater  than  the 
static  stresses,  due  largely  to  the  effect  of  unbalanced  locomo- 
tive wheels.  Where  a  large  amount  of  ballast  intervenes  be- 
tween the  load  and  the  structure  the  impact  is  doubtless  much 
less.  In  the  case  of  concrete  structures  the  great  mass  of  the 
concrete  undoubtedly  tends  to  reduce  the  effect  of  impact  and 
vibration,  or  to  localize  such  effect  more  than  in  a  steel  structure. 
The  conditions  involved  in  concrete  designing,  therefore,  are 
likely  to  be  favorable  as  regards  impact  and  may  permit  the 
use  of  lower  coefficients  than  are  used  for  steel  structures.  The 
proper  coefficient  to  use,  or  the  relation  between  live-  and  dead- 
load  working  stresses,  varies  much  under  different  conditions 
arid  must  be  left  to  the  judgment  of  the  designer,  or  to  formulas 
or  rules  prepared  especially  for  the  purpose.  Further  discussion 
of  this  question  will  not  be  undertaken  here. 

In  buildings  it  is  the  practice  in  steel  construction  to  use  a 
single  working  stress,  no  account  being  taken  directly  of  any 
special  effect  of  the  live  load.  Allowance  is  made  in  the  design 
of  large  girders  and  columns  which  receive  their  load  from 
large  areas  for  the  fact  that  such  large  areas,  especially  if  on  two 
or  more  floors,  are  seldom  or  never  loaded  to  the  extent  assumed 
for  smaller  areas .  This  allowance  varies  with  different  conditions, 


§  120.]  SAFE  WORKING  STRESSES.  169 

but  relates  solely  to  the  selection  of  the  amount  of  live  load 
rather  than  to  its  effect.  In  a  building,  when  heavily  loaded 
with  its  live  load,  the  portion  of  the  load  which  is  in  motion 
and  capable  of  producing  a  dynamic  effect  is  generally  but  a 
very  small  percentage  of  the  total  live  load.  In  most  cases, 
therefore,  in  building  construction  it  is  not  necessary  to  treat 
the  live-load  stresses  differently  from  the  dead-load  stresses, 
and  the  design  is  based  on  a  single  set  of  working  stresses. 
Special  cases  will  arise,  however,  where  the  dynamic  effect  of 
the  live  load  requires  consideration,  as,  for  example,  in  the  case 
of  floors  supporting  moving  machinery. 

Whatever  the  effect  of  live  load  may  be  it  can  more  readily  be 
taken  account  of  by  adding  to  the  resulting  live-load  stresses 
a  percentage  which,  in  the  judgment  of  the  engineer,  will  reduce 
them  to  their  dead-load  equivalent,  and  then  apply  a  single 
set  of  working  stresses,  or  factor  of  safety,  to  the  sum  of  the 
stresses.  The  discussion  of  working  stresses  in  the  following 
articles  will  relate  to  the  proper  basal  working  stress  for  dead 
load,  or  for  live  load  suitably  increased  for  impact. 

BEAMS. 

120.  Working  Formulas. — From  the  analysis  and  results  of 
experiments  discussed  in  preceding  chapters  there  would  appear 
to  be  no  good  reason  why  the  rational  formulas  as  developed  in 
Chapter  III  should  not  be  used  in  designing.  No  empirical 
formula  is  needed .  Furthermore,  in  the  judgment  of  the  authors, 
the  simple  formulas  based  on  the  straight- line  stress  variation 
should  be  used  for  purposes  of  design,  safe  working  stresses 
being  employed.  These  formulas  are  practically  correct  for 
such  working  stresses,  and  there  seems  to  be  no  more  reason 
to  use  formulas  designed  only  to  express  ultimate  strength  than 
there  is  in  the  case  of  wooden  or  cast-iron  beams  where  the 
conditions  are  similar.  It  is,  however,  desirable  that  the  work- 
ing stresses  be  selected  with  some  reference  to  ultimate  strength, 
although  with  principal  reference  to  elastic  strength. 


170  WORKING  STRESSES.  [Cn.  V, 

121.  Working  Stresses  in  Concrete  and  Steel.— The  strength 
of  a  beam  is  limited  usually  by : 

(a)  The  compressive  strength  of  the  concrete, 

(b)  The  elastic-limit  strength  of  the  steel,  or 

(c)  The  strength  of  the  beam  in  diagonal  tension. 

In  this  article  the  first  two  elements  only  will  be  considered. 

From  tests  relative  to  elastic  limit,  such  as  those  of  Bach 
and  Van  Ornum  (see  Chapters  II  and  IVy,  it  would  appear  that 
the  permanent  elastic  limit  of  concrete  is  from  50%  to  60% 
of  its  ultimate  strength  as  determined  in  the  usual  man- 
ner. If  a  factor  of  safety  of  two  be  applied  to  the  elastic -limit 
strength  to  provide  for  items  (a),  (b),  and  (c)  of  Art.  118,  we  will 
have  a  dead-load  basal  working  stress  of  25%  to  30%  of  the 
ultimate  strength.  Assuming  an  ultimate  strength  of  concrete 
in  cube  form  of  2000  to  2200  lbs/in2,  the  working  stress  \\ill 
then  be  from  500  to  600  lbs/in2. 

With  respect  to  the  steel  the  ultimate  strength  hardly 
comes  into  consideration;  its  elastic-limit  strength  is  nearly  its 
ultimate  strength  for  reinforcing  purposes.  So  far  as  safe 
stress  in  the  steel  is  concerened  a  working  stress  of  one-half  the 
elastic  limit  is  entirely  safe  as  a  dead-load  stress.  Let  it  be 
assumed  then,  for  illustration,  that  a  steel  is  used  having  an 
elastic-limit  strength  of  32,000  lbs/in2,  and  that  the  working 
stress  is  taken  at  16,000  lbs/in2.  Let  us  now  consider  the 
conditions  which  exist  in  a  beam  designed  with  the  above  work- 
ing stresses  when  subjected  to  an  increasing  load. 

For  a  load  which  produces  stresses  in  the  steel  or  concrete 
within  the  respective  elastic  limits,  the,  two  materials  are  indefi- 
nitely elastic  and  the  beam  is  entirely  secure ;  and  were  it  cer- 
tain that  the  stresses  would  under  no  conditions  exceed  these 
values  the  design  would  be  entirely  satisfactory.  Suppose, 
however,  that  under  emergency  conditions,  or  by  accident, 
the  stresses  pass  the  respective  elastic  limits,  it  will  be  noted 
that  as  regards  the  concrete  there  is  still  a  very  large  margin  of 
strength  (about  50%),  while  as  regards  the  steel  the  margin  is 
little  or  nothing.  Hence  the  beam  fails  through  excessive 


§  121.]  SAFE  WORKING  STRESSES.  171 

stresses  in  the  steel;  that  is,  the  ultimate  strength  of  the  beam 
is  limited  by  the  steel  to  a  value  much  below  its  strength  as 
determined  by  the  concrete.  It  follows  that  in  order  to  utilize, 
for  emergency  purposes,  the  strength  of  the  concrete  beyond 
its  elastic  limit,  the  working  stress  in  the  steel  must  be  selected 
so  as  to  give  the  desired  margin  of  strength  without  much 
exceeding  its  elastic  limit.  To  secure  these  conditions  in  the 
above  case,  the  working  stress  in  the  steel  would  have  to  be 
taken  at  about  8000  lbs/in2  or  a  material  of  higher  elastic 
limit  selected  in  order  to  support  an  ultimate  load  equal  to 
four  times  the  working  load.  The  concrete  would  be  able  to 
support  even  more  than  four  times  the  working  load,  since  at 
high  stresses  the  fibre  stress  no  longer  follows  the  straight-line 
law  of  stress  variation,  and  such  a  load  will  produce  a  fibre 
stress  considerably  less  than  2000  lbs/in2. 

Considering  the  fact  that  in  well-designed  beams  the  steel 
stress  at  failure  will  considerably  exceed  its  elastic  limit,  and 
considering  also  the  greater  reliability  and  uniformity  of  steel  as 
compared  to  concrete,  it  would  seem  that  a  working  stress  in 
the  steel  of  about  one-third  its  elastic  limit  would  correspond 
fairly  well  with  the  working  stress  in  the  concrete  here  suggested. 
With  such  values  for  working  stresses  the  beam  will  have  a 
factor  of  safety  as  regards  elastic  limit  of  about  two  (determined 
by  the  concrete),  and  as  regards  ultimate  strength  its  factor  of 
safety  will  be  at  least  five  relative  to  the  concrete  and  from 
three  to  four  relative  to  the  steel.  Its  elastic  limit  is  thus 
determined  by  the  concrete  and  its  ultimate  strength  by  the 
steel,  which  may  be  considered  as  satisfactory  conditions. 

The  working  stresses  in  the  steel  should  also  be  considered 
with  reference  to  its  distortion.  High  working  stresses  involve 
large  distortions,  and  hence  a  greater  degree  of  incipient  rupture 
in  the  concrete.  This  condition  is  probably  of  little  moment 
in  most  cases  so  far  as  it  concerns  undesirable  appearance  or 
exposure  of  steel  to  corrosion,  but  is  of  importance  with  reference 
to  its  effect  on  diagonal  tensile  stresses  as  explained  in  Art.  109. 
Low  unit  stresses  in  the  steel  are  greatly  to  be  preferred  on  this 


172  WORKING  STRESSES.  [Cn.  V. 

account.  It  will  also  be  shown  in  Art.  133  that  very  little  is  to 
be  gained  in  economy  by  using  stresses  higher  than  about  12,000 
lbs/in2.  Considering  this  fact  and  the  objections  to  high 
stresses  above  mentioned,  it  would  seem  that  a  stress  of  15,000 
lbs/in2  should  be  considered  the  maximum  desirable  value 
irrespective  of  the  quality  of  the  steel  used.  A  lower  value  is 
strongly  to  be  recommended.  Finally,  as  the  result  of  this 
analysis  we  may  conclude  that  the  basal  working  stress  in  the 
steel  should  not  exceed  about  one-third  its  elastic  limit  nor 
exceed  15,000  lbs/in2. 

122.  Quality  of  Steel. — As  stated  in  Art.  34,  there  exists 
considerable  difference  of  opinion  as  to  the  quality  of  steel  to 
be  desired,  especially  with  reference  to  the  use  of  soft  or  hard 
material,  or  steel  with  low  or  high  elastic  limits.     Certainly  a 
material  as  hard  as  that  formerly  denominated  "hard  bridge 
steel"  is  entirely  suitable  for  reinforced  construction.     Such 
material  has  an  elastic  limit  of  about  40,000  Ibs./in  2.     Much 
material  has  been  used  of  an  elastic  limit  of  45,000  to  50,000 
lbs/in2  and  even  higher,  but  a  value  beyond  this  is  not  to  be 
desired.    An  elastic  limit  of  45,000  lbs/in2  is  three  times  the 
working  stress  of  15,000  lbs/in2.    The  use  of  a  steel  with  an 
elastic  limit  higher  than  this  is  unnecessary  and  is  of  doubtful 
wisdom,  as  the  ductility  of  a  higher  steel  of  the  usual  quality  is 
not  high.    The  authors  would  suggest  a  material  of  the  quality 
employed  for  buildings  with    an    elastic    limit    of    35-40,000 
lbs/in2  and  working  stresses  of  12,000  to  14,000  lbs/in2. 

123.  Bond  Stress. — The  factor  of  safety  with  reference  to 
the  slipping  of  the  rods  should  be  at  least  5,  since  the  strength 
of  a  beam  should  not  be  limited  by  the  strength  of  bond.    From 
the  data  of  Chapter  IV,  we  may  take  the  ultimate  bond  strength 
of  plain  steel  at  from  250  to  400  lbs/in2.    A  working  stress 
of  from  50  to  75  lbs/in2  would  therefore  be  suitable.    With  a 
working  bond  stress  of  60,  say,  and  a  tensile  unit  stress  of  15,000 
a  round  bar  will  need  to  be  embedded  a  length  of  15, 000/4  X  60  = 
62.5  diameters  to  develop  its  full  strength.     In  the  case  of  large 
bars  of  1"  to  1J"  in  diameter  this  length  is  very  considerable 


§  125.]  WEB  REINFORCEMENT.  173 

and  for  short  beams  may  be  difficult  to  secure.  The  deformed 
bar,  or  the  anchored  bar,  is  of  especial  value  under  these  condi- 
tions. 

For  deformed  bars  the  safe  working  stress  may  be  taken 
at  about  100  lbs/in2,  thus  requiring  a  length  of  embedment  of 
about  15,000  /4  X 100  =  37.5  diameters. 

124.  Shearing  Stresses. — From   the    results   discussed   in 
Chapter  IV  the  ultimate  shearing  strength  of  a  beam  having 
no  web  reinforcement   may  be  taken   at    about   100  lbs/in2, 
calculated  as  average  shearing  stress  on  the  cross-section.    In- 
asmuch as  a  failure  due  to  high  shearing  stresses  is  apt  to  be 
sudden,  the  factor  of  safety  should  be  fully  as  great  as  that 
with  reference  to  the  compressive  strength  of  the  concrete. 
This  gives  a  working  stress  of  25  to  30  lbs/in2.     For   beams 
in  which  the  web  is  well  reinforced  the  working  stresses  may  be 
made  3  or  4  times  as  great,  or  from  75  to  100  lbs/in2. 

125.  Calculation  of  Web  Reinforcement. — Sufficient  exper- 
imental work  has  not  been  done  to  enable  the  proportioning  of 
web  reinforcement  to  be  done  with  any  degree  of  exactness. 
However,  a  rough  estimate  of  the  requirements    can  be  deter- 
mined on  rational  grounds.    The  tests  already  quoted  in  Chap- 
ter IV  indicate  that  beams  with  horizontal  bars  only  cannot  be 
stressed  safely  beyond  about  30  lbs/in2  average  shearing  stress, 
the  strength  depending  on  the  quality  of  concrete  and  the  unit 
stresses  adopted  for  the  horizontal  steel.     In  practice  it  will 
rarely  happen  that  a  beam  need  carry  more  than  100  lbs/in2 
average  shearing  stress,  and  tests  of  the  best  work  indicate  that 
this  should  be  about  the  maximum  limit.    In  such  a  case  if 
the  concrete  be  assumed  to  carry  30  lbs/in2  the   steel  must 
carry  the  remainder  at  a  working  stress  of  say  15,000  lbs/in2. 
If  the  area  of  cross-section  of  beam  be  bd,  then  the  shear  to  be 
carried  is  70frd;    and  as  the  tendency  to  rupture  is  on  a  line 
inclined  at  45°,  this  shear  may  be  considered  as  the  load  to  be 
carried  by  the  web  reinforcement  in  a  length  equal  to  the  depth 
d.    The  necessary  steel   area   is   therefore   A  =  706J ,/15,000  = 

7bd,  or   .47%.    The  area  of  the  horizontal  reinforcement 


174  WORKING  STRESSES.  [On.  V. 

should  not  be  taken  into  account,  as  is  sometimes  done.  Where 
the  shear  is  large,  as  in  the  case  assumed,  the  stirrups  or  other 
reinforcing  members  should  be  placed  not  farther  apart  than 
?d,  and  the  sectional  area  of  each  may  therefore  be  taken  =  .23% 
of  the  cross-section.  For  example,  a  beam  8"Xl2"  would 
require  stirrups  6  in.  apart  and  each  of  a  cross-section  of  .23%  X 
96  =  .22  pi.2.  This  would  be  equivalent  to  a  f-in.  stirrup  in  a 
single  loop  or  a  J-in.  stirrup  in  a  double  loop.  Close  spacing 
is  of  more  importance  even  than  size,  but  high  working  stresses 
are  undesirable,  as  they  permit  too  great  distortions,  with 
resulting  minute  cracks  in  the  concrete.  The  value  of  a  stirrup 
or  the  bent  end  of  a  bar  is  also  limited  by  its  safe  bond  strength. 
Inclined  rods  are  almost  necessarily  of  too  large  size  and  too  far 
apart  to  be  effective  without  some  stirrups,  but  the  two  com- 
bined, using  fewer  stirrups,  is  an  effective  combination.  Where 
the  shear  in  the  beam  is  less  than  the  allowed  value,  web  rein- 
forcement may  be  omitted. 

126.  Spacing  of  Bars. — In  rectangular  or  T-beams  the  spac- 
ing of  bars  is  important;  in  T-beams  this  consideration  will 
largely  control  the  width  of  the  beam.  The  requirement  in 
general  as  to  spacing  is  that  the  amount  of  concrete  left 
between  the  bars  must  be  sufficient  to  transmit  to  the  upper 
part  of  the  beam  the  stress  which  the  bars  give  over  to  the 
concrete  below  them.  If  the  bars  are  circular  it  may  be  assumed 
that  one-half  of  the  stress  in  them  is  given  over  to  the  concrete 
below,  hence  the  strength  of  the  concrete  on  a  longitudinal 
section  through  the  center  plane  of  the  bars  must  equal  one- 
half  of  the  stress  in  the  bars.  If  the  shearing  stress  be  taken 
as  equal  only  to  the  bond  stress  then  the  clear  space  between 
bars  must  be  one-half  the  circumference  of  a  bar,  or  1.57 
diameters.  In  the  sense  here  employed  the  shearing  strength 
is  at  least  twice  the  bond  strength  for  smooth  rods,  so  a  clear 
spacing  of  less  than  one  diameter  is  sufficient  from  this  stand- 
point. In  the  case  of  square  bars,  on  the  same  basis,  the  clear 
spacing  would  need  to  be  1J  diameters  if  the  bars  are  placed 
with  sides  vertical,  or  one  diameter  is  placed  with  sides  diagonal. 


§  127.]  ECONOMICAL  WORKING  STRESSES.  175 

But  in  addition  to  the  shearing  stresses  there  is  likely  to  be 
developed  more  or  less  tension  in  the  concrete  surrounding  the 
rods,  so  that  there  should  be  left  ample  areas  of  concrete  between 
them,  especially  towards  the  end  where  the  bond  stresses  are 
large.  A  minimum  clear  spacing  of  at  least  1|  diameters  should 
be  provided,  with  an  equal  distance  between  the  outside  rod 
and  the  surface  of  the  beam.  Where  some  of  the  rods  are  bent 
up  the  spacing  can  readily  be  made  more  liberal  towards  the 
end  of  the  beam. 

Liberal  spacing,  or  large  net  section  of  concrete,  favors  large 
rods  and  few  in  number;  good  bond  strength  without  waste  of 
material  favors  small  rods.  If  bent  rods  are  to  be  used  for  web 
reinforcement,  then  numerous  small  rods  are  also  advantageous. 
If  the  bond  strength  is  not  in  question,  or  can  easily  be  taken 
care  of,  then  large  rods  are  desirable,  but  more  stirrups  or  other 
secondary  reinforcement  may  be  needed  than  where  small  rods 
are  used. 

127.  Economical  Proportions  and  Working  Stresses. — For 
given  unit  prices,  the  cost  of  concrete  beams  per  unit  of  resisting 
moment  will  vary  with  the  proportions  adopted  for  breadth  and 
depth,  and  with  the  working  stresses  employed.  Because  of 
the  mutual  relations  between  the  concrete  and  steel  it  may 
happen  that  the  maximum  economy  of  construction  may  be 
obtained  by  using  less  than  the  allowable  working  stresses  in 
one  or  the  other  of  the  two  materials.  It  will  therefore  be 
useful  to  investigate  the  effect  on  cost  of  variations  in  propor- 
tions and  in  the  working  stresses. 

Consider  a  portion  of  a  rectangular  beam  one  unit  in  length. 

Let    c  =  cost  of  concrete  per  unit  volume; 

r  =  ratio  of  cost  of  steel  to  cost  of  concrete  per  unit 

volume; 

p= ratio  of  steel  area  to  concrete  area; 
(7  =  cost  of  beam  per  unit  length. 

Then  C  =  c(bd+rpbd)  =cbd(l  +  rp).     ....    .     (1) 

From  Art.  59  we  have  bd?  =  M/R,  in  which  M  =  bending 


176  WORKING  STRESSES.  [Cn.  V. 

moment  and  R  =  coefficient  of  strength  of  the  beam,  depending 
in  value  only  upon  /s,  /c,  and  n.  From  this  we  may  write 
bd=M/*Rd,  bd=VMb/R,  and  bd=V(b/d)(M2/R2),  whence  we 
derive  the  three  expressions  for  cost : 

M 


^,  (3) 


and 


128.  General  Effect  of  Varying  Proportions. — Since  the  values 
of  R  and  p  depend  only  on  /.,  fc  and  n  we  note  from  (2)  that  the 
cost  of  a  rectangular  beam  to  support  a  given  moment,  M ,  varies 
inversely  with  the  depth;  and  from  (3)  that  the  cost  varies 
directly  with  Vbreadth;  and  finally,  from  (4)  that  it  varies 
with  the  cube  root  of  the  ratio  of  breadth  to  depth.  In  all  cases 
it  is  assumed  that  the  two  dimensions  are  made  to  correspond 
with  each  other  as  calculated  from  the  selected  values  of  /„ 
and  /c.  It  follows  from  (2)  that  with  given  values  of  f8  and  fc 
the  deeper  the  beam  the.  less  the  cost,  so  long  as  b  can  be  reduced 
accordingly.  The  depth  will,  however,  be  limited  in  various 
ways.  It  may  be  limited  by  the  requirement  of  shearing  stress 
fixing  the  value  of  bd,  or  it  may  be  limited  by  the  head  room 
required,  or  it  may  practically  be  limited  by  the  fact  that  a 
certain  breadth  is  necessary  to  give  a  convenient  and  proper 
covering  of  the  steel  reinforcement  or  to  give  a  beam  of  satis- 
factory proportions.  In  the  construction  of  continuous  surfaces, 
such  as  floor  slabs,  the  case  is  one  of  fixed  width,  since  the  width 
of  beam  to  carry  the  load  coming  upon  a  strip  one  foot  wide 
is  also  one  foot.  We  may  then  consider  four  cases  according 
to  the  particular  feature  of  the  design  which  is  the  controlling 
element.  These  cases  are: 

(a)  When  the  area  of  cross-section  is  determined  by  the 

shear ; 

(b)  When  the  depth  of  the  beam  is  fixed: 


§  130.]  ECONOMICAL  WORKING  STRESSES.  177 

(c)  When  the  width  of  the  beam  is  fixed; 

(d)  When  the  ratio  of  width  to  depth  is  fixed. 

129.  (a)  The   Area   of  Cross-section  is  Determined  by  the 
Shear. — A  given  value  for  shearing  stress  requires  a  fixed  value  of 
bd,  but  the  requirement  for  bending  moment  is  that  bd  =  M/Rd', 
hence  if  a  beam  is  designed  for  moment  alone  the  area  bd  will  be 
less  the  deeper  the  beam.    Theoretically,  therefore,  for  a  given 
value  of  R  the  maximum  depth  permissible  is  that  for  which  the 
resulting  area  bd  is  just  large  enough  to  carry  the  shear.    If  V  is 
the  total  shear  and  v'  is  the  permissible  shearing  stress,  then  bd  = 
V/v'.    Also  bd  =  M/Rd.     Hence  for  equal  strength  M/Rd  =  V/v' 
and  therefore 

d  =  Mv'/RV     . (5) 

and  b  =  V/v'd (6) 

These  equations  give  the  dimensions  of  a  beam  which  will 
be  of  just  the  required  strength  in  moment  and  shear.  It  re- 
mains to  be  determined,  however,  whether  a  still  greater  depth 
will  result  in  greater  economy. 

If  a  greater  depth  be  used,  bd  must  remain  constant;  hence 
bd2  will  be  increased  and  the  concrete  stress,  fc,  decreased. 
Reference  to  Plate  III,  p.  215,  shows  that  with  constant  /„  a  de- 
crease in  the  value  of  fc  permits  the  use  of  a  smaller  percentage 
of  steel.  Hence  with  increasing  depth  and  constant  bd  (or 
volume  of  concrete),  the  amount  of  steel  will  be  reduced,  and 
therefore  the  cost.  The  proportions  of  the  beam  will  therefore 
not  be  determined  by  the  shear  excepting  as  to  minimum  cross- 
section. 

130.  (b)  The  Depth  of  the  Beam  is  Fixed. — From  eq.  (2)  it 
is  seen  that  for  given  values  of  M  and  d  the  cost  varies  with 
(1  +  rp)/R.    Now  p  and  R  depend  only  upon  the  working  stresses 
f8  and  fc  (n  being  constant),  hence  it  will  be  convenient  to 
determine  the  variation  in  cost  due  to  variation  in  /8  and  fC9 
assuming  certain  values  for  r.    Results  of  this  analysis  are 
shown  in  Fig.  58  for  values  of  r  of  60  and  80  and  for  various 
values  of  fs  and  fc.    The  results  are  very  instructive  and  show 


178 


10000 


.024 
.022 
.020 
.018 
.016 
JD14 
.0.12 

,028 
.026 
.024 
.022 
.020 
.018 
.016 

,014 


.0, 


WORKING  STRESSES.  [Cn.  V. 

12000  14000  16000  18000  20000 


fs 


600 


80 


.0000  12000  14000  JI6000  18000 

FIG.  58.— Relative  Cost  for  Fixed  Depth. 


20000 


§  133.]  ECONOMICAL  WORKING  STRESSES.  179 

that  for  values  of  fc  of  500  or  600  lbs/in2  no  economy  is  secured 
by  using  values  of  fs  greater  than  12-14,000  lbs/in2.  For  larger 
values  of  r  or  of  fc,  higher  values  can  economically  be  used  for  /s, 
but  a  value  of  80  for  r  is  not  likely  to  be  greatly  exceeded.  If  the 
cost  of  concrete  be  as  low  as  20  c.  per  cu.  ft.  the  corresponding 
cost  of  steel  would  be  $16.00  per  cu.  ft.,  or  3.2  c.  per  pound. 
This  is  a  low  cost  of  concrete  and  a  high  cost  of  steel.  The  dia- 
gram shows  tha  the  cost  is  decreased  by  increased  values  of  fc. 

131.  (c)  The  Width  of  the  Beam  is  Fixed. — From  eq.  (3) 
the  cost  for  given  values  of  M  and  b  varies  with  (l+rp)/VR. 
Fig.  59  represents  this  quantity  plotted  for  various  values  of 
fs  and  /c.    Comparing  this  with  Fig.  58  it  is  seen  that  somewhat 
higher  values  of  /,  are  warranted,  but  it  is  evident  that  the 
gain  in  economy  is  very  small  for  values  above  16,000  lbs/in2, 
except  where  the  steel  is  very  expensive  and  the  concrete  cheap. 

132.  (d)  The  Ratio  of  Width  to  Depth  is  Fixed. — It  is  often 
desired  to  secure  approximately  a  certain  given  ratio  of  breadth 
to  depth.    In  this  case  we  find  from  eq.  (4)  that  the  cost  will 
vary  with  (l  +  rp)/R$.    Fig.   60   represents   this   quantity  for 
various  values  of  fs  and  fc.    It  is  seen  that  the  most  economical 
values  will  lie  between  those  of  cases  (b)  and  (c). 

133.  Floor  Slabs  with  Weight  of  Concrete  Eliminated. — In 
all  the  foregoing  discussion  the  moment  to  be    resisted    has 
included  that  due  to  the  weight  of  the  beam  itself.     For  large 
beams  and  girders  this  is  unimportant  in  this  connection,  but 
with  floor  slabs,  where  the  external  load  is  small,  the  weight  of 
the  material  itself  modifies  the  results  to  a  large  extent.    General 
results  cannot  be  presented  for  all  cases,  but  the  analysis  will  be 
given  for  a  single  case  representing  ordinary  conditions.    A 
span  length  of    10  ft.  has  been  taken  and  a  net  floor  load  of 
150  lbs/ft2.    Then  from  Table  No.  21,  Chap.  VI,  the  required 
cross-section  and  amount  of  steel  has  been  determined  for  vari- 
ous values  of  f8  and  fc.    The  relative  cost  per  unit  floor  area  has 
been  calculated  for  values  of  r  of  40,  60,  80,  and  100  and  the  re- 
sults plotted  in  Fig.  61.     Comparing  these  results  with  those  of 
Fig.  59,  where  the  weight  of  the  beam  has  not  been  deducted, 


180 


10000 
.1800 


.1750 
.1700 
.1650 
.1600 
.1550 
.1500 
.1450 


.1400 

.2000 


.1950 


1.1900 

s 

K.1850 
.1800 
.1750 
.1700 
.1650 
.1600 
.1550 
.1500 
.1450 


WORKING  STRESSES.  [Cn.  V. 

12000  14000  16000  18000  20000 


J400 


500 


f. 


»•— 80 


.1800 
.1750 
.1700 
.1650 
.1600 
.1550 
.1500 
.1450 
.1*400 
.1350 


.1300 
.1900 


.1850 
.1800 
.1750 
.17.00 
.1650 

.1600 
,f550 


10000       12000       14000       16000       18000 

FIG.  59. — Relative  Cost  for  Fixed  Width. 


.1500 
.1-450 

1 .1400 

20000 


§  133.]  ECONOMICAL  WORKING  STRESSES. 

10000        12000        WOOO        16000        18000 


181 


& 


.100 
.095 
J090 
J085 

.080 
.075 
.070 
.065 

.060 
i 
i 
i 
[ 
)  .055 

;  .105 

;100 
.095 
.090 
.085 
.080 
.075 
.070 
.065 
*060 


f. 


=  600 


r-60 


20000 


100 


095 


090 


085 


080 


075 


,070> 


OG5 


oca 


055 
105 


100 


095 


090 


075 


070 


OG5 


060 


10000        12000        14000        16000        18000        SOOOO 

FIG.  60. — Relative  Cost  for  Fixed  Ratio,  Breadth  to  Depth 


WORKING  STRESSES. 


jpm 


=  500 


600 


700 


400 


/c=500 


=600 


700 


/S 


40 


60 


Vl2000  13000  14000  15000  16000  17000  18000 

FIG.  61a.— Relative  Cost  for  Fixed  Breadth,  Weight  of  Beam  Deducted. 


§  135.] 


ECONOMICAL  WORKING  STRESSES. 


183 


£ 


12 
1.05 

1.00 
,95 

000              13000                14000               15000                16000                17000                180 

00 
1.05 

1.00 
.95 

fs 
=-400— 

r 

=  80 

.90 

^•^ 

*•  

1  —  -»» 

*•  •, 

/ 

.90 

.85 

1  
•^^ 

—  •  —  , 

^*-  —  , 

**  

-*•  —  . 

•—  — 

-       *LL 

~ 

• 

—  ^—  — 

-^^—  —  ^~ 

.85 

.80 

.75 

. 

1.10 
1.05 
1.00 

"—•—-«. 

—  •  — 

^ 

^£ 

^ 

~             — 
—  . 

— 
•  

—      - 

•^    — 

•V^^^^M 

—  '       — 

—  —  — 

— 

.80 
.75 

1.10 
1.05 
1.00 

1  

—     — 

—     — 

~i    — 

- 

• 

=3fe 

-  4OO 

--^ 

•*-  

T, 

=100 

.95 

^^ 

•^^^ 

^**1—  *» 

^-^ 

— 

.95 

.90 

^^ 

^^^*», 

-^ 

"  

!^^ 

^  —  . 
^^; 

^*  — 
•^-^ 

-"-   ^ 
^= 

"^—  

500 

—     — 

~           _ 

-—  ^_^ 

-^—  —  ^~— 

.90 

^600 

.85 

.75 
70 

«^£ 

=^: 

—  -^ 
, 

""  —  1 

•*•    ^ 

"•        fc. 
—      -. 

.85 
.80 
.75 
70 

fs 

13000  13000  14000  15000  16000  17000  18000 

FIG.  616.— Relative  Cost  for  Fixed  Breadth,  Weight  of  Beam  Deducted. 


184  WORKING  STRESSES.  [Cn.  V, 

it  is  seen  that  the  economical  values  of  fa  are  considerably  less. 
For  values  of  r  not  exceeding  60  and  for  fc  not  exceeding  500 
there  is  no  reason  for  using  a  value  for  fs  higher  than  14,000 
lbs/in2.  For  other  spans  and  floor  loads  the  results  will  be 
somewhat  different,  but  the  variation  will  not  be  great.  Larger 
floor  loads  and  shorter  spans  will  give  results  more  nearly  ap- 
proaching those  of  Fig.  59;  smaller  loads  and  longer  spans  v>ill 
tend  in  the  opposite  direction. 

Percentages  of  steel  corresponding  to  any  particular  values 
of  }c  and  fa  are  given  by  reference  to  Plate  III,  p.  215. 

134.  Effect  of  Overlapping  Bars. — In  most  cases  the  rein- 
forcing bars  of  slabs  are  made  to  overlap  more  or  less;   where 
negative  moment  over  the  beams  is  taken  care  of  this  over- 
lapping may  be  25  to  30  per  cent.    To  take  account  of  this 
in  using  the  equations  or  diagrams  of  the  preceding  articles, 
the  most  convenient  method  is  to  increase  the  unit  cost  of  steel, 
or  the  ratio  r,  by  the  same  percentage  that  measures  the  over- 
lap of  the  steel. 

135.  T-Beams.— In  the  case    of  T-beams,  the  slab  forms 
practically  all  the  compressive  area,  but  does  not  enter  into 
the  cost  of  the  beam.    Using  the  approximate  formula,  eq.  (7), 
of  Art.  74  the  area  of  the  steel  is  equal  to  M/fs(d'+  %t),  in  which 
d'  is  the  depth  of  beam  below  the  slab.    The  cost  is  then 


From  this  expression  it  is  evident  that  the  cost  will  decrease 
with  increased  values  of  fs  under  all  conditions,  and  that  with 
a  fixed  value  of  b'd'  the  cost  decreases  with  increase  in  depth. 
If  df  is  fixed  then  the  cost  will  be  a  minimum  when  &'  is  made 
as  small  as  possible,  and  its  value  will  then  be  determined  by 
the  shearing  stress  or  by  the  space  required  for  the  bars.  If 
the  value  of  V  is  assumed  as  fixed,  then  there  is  a  definite 
value  of  d'  which  will  give  minimum  cost.  Considering  d'  as 
variable  and  &'  as  constant  we  find  by  differentiation  that  for 


§  136.]  COLUMNS.  185 

. 
minimum  cost  the  value  of  d'  is  given  by  the  equation 


d'  +  t/2=VrM/fsb' (2) 

From  this  expression  the  best  depth  for  various  assumed  widths 
can  readily  be  determined  and  the  desirable  proportions  finally 
selected. 

T-beams  should  not  be  made  too  deep  in  proportion  to 
width,  as  such  forms  are  relatively  weak  at  the  junction  of  stem 
and  flange.  All  re-entrant  angles  in  rigid  material  such  as 
concrete  are  points  of  weakness  and  such  angles  should  therefore 
be  modified  by  curved  lines  or  made  obtuse  by  sloping  the  sides 
of  the  beam.  A  width  of  beam  sufficient  to  carry  the  shear  and 
to  give  plenty  of  space  for  the  bars  is  usually  ample.  The 
maximum  desirable  ratio  of  depth  to  width  may  be  taken  at 
about  two  for  small  beams  up  to  three  or  four  for  very  large 
and  massive  work.  Depths  are  often  determined  by  available 
head  room.  Beams  of  excessive  depths  are  objectionable  as 
being  more  difficult  and  troublesome  to  reinforce  properly; 
the  cost  of  web  reinforcement  also  becomes  relatively  greater. 

COLUMNS. 

136.  Working  Stresses. — The  working  stresses  for  columns 
should  represent  at  least  as  great  a  factor  of  safety  as  for  beams. 
The  experiments  noted  in  Chapter  IV  indicate  that  concrete  in 
the  form  of  a  column  is  not  as  strong  as  in  the  cube  or  beam 
form.  A  value  of  about  1600  lbs/in2  for  1:2:4  plain  con- 
crete would  seem  to  be  a  fair  value  for  ultimate  strength,  and 
applying  a  factor  of  safety  of  four  gives  a  working  stress  of 
400  lbs/in2. 

The  working  stress  in  the  steel  is  a  function  of  the  working 
stress  in  the  concrete  and  the  ratio,  n,  of  the  moduli  of  elas- 
ticity of  the  two  materials.  If  this  ratio  is  taken  at  12,  then 
the  working  stress  in  the  steel  must  be,  in  the  above  case, 
12  X  400  =  4800  lbs/in2.  Under  working  loads  the  steel  is  there- 
fore stressed  only  to  a  very  low  value. 


186 


WORKING  STRESSES. 


[CH.  V, 


Let  us  consider  the  variation  in  the  stresses  in  a  column 
subjected  to  increasing  loads.  Fig.  62  represents  a  stress-strain 
diagram  of  1:2 -A  concrete  in  compression.  The  modulus  up  to 
500-600  lbs/in2  is,  say,  2,500,000;  the  modulus  at  rupture 
(ratio  of  stress  to  total  deformation)  is  perhaps  only  1,000,000, 
In  Fig.  63  let  absciss*  represent  unit  stress  in  concrete  in  the 
given  column  up  to  1600  lbs/in2.  Let  the  ordinates  above 
the  axis  represent  the  total  stress  in  the  steel  corresponding 
to  various  unit  stresses  in  the  concrete.  For  low  values  of  fc 
the  value  of  n  is  12  and  fs  =  12/c.  For  higher  values  of  fc  the 
value  of  n  increases  until  at  the  maximum  of  1600  lbs/in2, 
n  =  30  and  /s  =  30/c,  or  48,000  lbs/in2  The  curve  OAB  rep- 

B, 


,400 


Unit  stress  in  Concrete 
,800  ,1200        ,1600 


FIG.  62, 


FIG.  63. 


resents  the  variation  in  the  total  stress  fsA8.  The  total  stress 
on  the  concrete,  fcAc,  may  likewise  be  conveniently  represented 
by  ordinates  from  OX  to  a  straight  line  OC,  the  scale  below 
OX  representing  concrete  stress.  Then  for  any  load  causing 
a  particular  unit  stress  in  the  concrete,  the  total  ordinate 
between  the  lines  OB  and  OC  will  represent  this  load.  From  this 
it  is  plain  that  with  increasing  loads  the  steel  receives  a  greater 
proportionate  stress,  the  variation  in  the  amount  carried  by  the 
steel  depending  on  the  variation  in  the  value  of  n.  It  is  also 
evident  from  this  diagram  that  the  ultimate  load  on  the  column 
is  much  greater  than  four  times  the  load  which  produces  the 


§  136.]  COLUMNS.  187 

stress  of  400  lbs/in2  in  the  concrete.  Hence  if  the  working  stress 
in  the  concrete  is  based  on  a  factor  of  safety  of  four  relative 
to  plain  concrete,  then  the  factor  of  safety  of  the  reinforced 
column  will  be  greater  than  four.  The  case  is  somewhat  similar 
to  that  of  the  beam.  Obviously  the  total  load  increases  more 
rapidly  than  the  value  of  the  stress  /c,  the  exact  rate  depend- 
ing on  the  relative  amount  of  steel  and  the  variation  in  n. 

For  purpose  of  calculation  the  formula  of  Art.  95,  Chap.  Ill, 
is  convenient.  From  this  the  total  load  for  a  reinforced  col- 
umn is 

.....    (1) 


in  which  p=  steel  ratio  and  A  =  total  area. 

To  give  a  numerical  illustration,  let  p  =  l%,  /c=400,  and 
n  =  12;  then  £"=4004X1.11.  For  /c  =  1600  and  n  =  30,  P'  = 
1600  A  X  1.29.  The  second  value  is  4.65  times  the  former  value, 
thus  giving  to  the  column  a  factor  of  safety  relative  to  rupture 
of  4.65. 

One  result  of  the  increased  stress  taken  by  the  steel  under 
increasing  loads  is  that  columns  containing  different  amounts 
of  steel  will  have  different  factors  of  safety  relative  to  ultimate 
strength,  even  though  calculated  for  the  same  working  stresses. 
For  example,  consider  a  series  of  columns  in  which  p  =  1,  2,  3,  4, 
and  5  per  cent,  and  all  having  the  same  area  A.  Their  relative 
strengths  at  a  value  of  fc  =  400  and  n  =  12  will  be  as  represented 
by  the  quantities  1.11,  1.22,  1.33,  1.44,  1.55.  At  ultimate 
load,  determined  by  the  value  of  1600  for  fc  and  with  n  =  30, 
the  relative  strengths  will  be  as  4x  1.29,  4X  1.58,  X  1.89,  4x2.16, 
and  4X2.45.  Dividing  this  series  by  the  former  series  we  have 
the  factors  of  safety  as  follows:  4.65,  5.18,  5.68,  6.00,  and  6.32. 
The  column  having  5%  of  steel  has  therefore  a  factor  of  safety 
1.36  times  as  great  as  the  column  with  1%  of  steel. 

In  order  to  secure  a  more  uniform  factor  of  safety,  and  to 
take  some  account  of  the  fact  that  under  increasing  loads  the 
steel  receives  an  increasing  proportion,  it  would  seem  desirable 
to  use  a  value  of  n  in  the  calculations  somewhat  larger  than 


188  WORKING  STRESSES.  [Cn.  V. 

that  which  is  obtained  by  taking  a  value  of  Ec  corresponding 
to  very  low  stresses.  A  value  of  15  or  even  20  might  well  be 
taken  for  1:2:4  concrete.  On  this  basis  the  calculations  will 
give  a  little  more  stress  in  the  steel  than  actually  exists  under 
the  usual  working  loads,  but  will  give  too  small  stress  under 
ultimate  loads.  In  the  case  of  hooped  columns  it  is  not  yet 
clear  just  what  weight  should  be  given  to -this  form  of  rein- 
forcement. Until  further  tests  are  available  it  would  hardly 
seem  wise  to  assign  any  greatly  increased  value  to  this  form 
over  longitudinal  metal,  although  such  a  column  undoubtedly 
is  capable  of  greater  deformation,  or  possesses  greater  "tough- 


ness". 


137.  Economy  in  the  Use  of  Reinforced  Columns.— From 
eq.  (1)  we  see  that  with  a  value  of  n  =  15,  the  use  of  each  1% 
of  steel  adds  14%  to  the  strength  of  a  column.  If  the  ratio 
of  cost  of  steel  to  cost  of  concrete  per  unit  volume  be  50,  then 
the  increased  cost  of  a  column  with  1%  of  steel  will  be 
50X1%  =50%.  The  gain  in  strength  being  only  14%,  the 
relative  economy  of  the  reinforced  column  is  only  iM  =  76% 
that  of  the  plain  concrete.  Again,  take  a  very  strong  mixture, 
such  as  1 : 1  mortar,  whose  working  stress  may  possibly  be  taken 
as  high  as  800  lbs/in2.  Such  a  mortar  will  cost  perhaps  $12.00 
per  cu.  yd.  (not  including  forms,  etc.)  or  45  c.  per  cu.  ft. 
Placing  steel  at  the  low  value  of  2  c.  per  lb.,  the  cost  ratio 
becomes  22.5.  Such  concrete  will  have  a  value  of  Ec  of  at 
least  3,000,000,  giving  71  =  10.  Hence  1%  reinforcement  will 
add  9%  to  the  strength  and  22.5%  to  the  cost.  If  a  cheap 
concrete  be  taken  with  a  low  modulus  the  steel  will  add  a  larger 
percentage  of  strength,  but  at  the  same  time  a  much  greater 
percentage  of  cost.  Another  way  of  considering  this  question 
is  from  the  standpoint  of  working  stresses  in  the  steel,  which 
can  scarcely  be  greater  than  8000  lbs/in2  under  working  con- 
ditions. The  cost  to  carry  a  given  load  on  the  steel  is  then 
(P/8000)Xcost  of  steel.  With  500  lbs/in2  working  stress  in 
concrete  the  cost  to  carry  the  load  on  the  concrete  is  (P/500)  X 
cost  of  concrete.  The  relative  cost  of  the  two  materials  is  then 


§  138.]  COLUMNS.  189 

(500/8000)  Xcost  ratio  of  steel  to  concrete.  If  this  ratio =50 
then  the  relative  cost = 25,000/8000  =  3J;  that  is,  the  steel  is 
3J  times  as  costly  as  an  equivalent  amount  of  concrete. 

The  above  analysis  shows  that  from  the  standpoint  of 
theoretical  economy  the  use  of  steel  in  columns  is  undesirable, 
and  were  this  the  only  consideration  it  would  not  be  used,  at 
least  in  the  form  discussed.  While  no  economy  can  be  figured 
for  the  use  of  steel  in  columns  it  is  by  no  means  valueless.  In 
practice,  columns  are  subjected  to  bending  moments  uncertain 
in  amount,  but  for  which  something  more  than  plain  concrete  is 
desired,  especially  where  the  column  is  of  considerable  length. 
It  is  in  such  columns  that  tensile  stresses  are  most  apt  to  occur 
and  where  steel  is  most  needed.  Furthermore,  steel  is  a  more 
reliable  material  than  concrete,  and  in  small  sections  where  the 
danger  of  weak  or  imperfect  spots  in  the  concrete  is  greatest, 
steel  reinforcement  is  of  great  value  in  producing  a  more  reliable 
structure.  Then,  again,  great  strength  may  be  desired  from 
small  sections  in  order  to  save  space,  in  which  case  steel  may  be 
used.  In  very  large  (relatively  short)  columns  little  is  to  be 
feared  from  bending  stresses,  as  in  such  a  case  no  resultant 
tensile  stress  is  likely  to  occur.  In  general  the  above  discussion 
shows  that  where  the  concrete  may  be  assumed  to  carry  its 
share  of  the  load  the  amount  of  longitudinal  steel  should  be 
made  small,  the  amount  preferably  increasing  with  increasing 
ratio  of  length  to  diameter. 

138.  Use  of  Steel  of  High  Elastic  Limit. — In  order  that  the 
elastic  limit  of  the  steel  may  not  limit  the  strength  of  the  column 
it  must  be  somewhat  high.  If  the  ultimate  strength  of  the 
crete  be  1600  lbs/in2  and  for  this  stress  n  =  30,  then  at  rupture 
the  stress  in  the  steel  will  be  48,000  lbs/in2.  For  weaker  and 
stronger  concretes  the  product  of  fc  and  n  will  not  be  greatly 
different,  as  the  value  of  Ec  varies  with  the  strength  of  the 
concrete.  For  columns,  therefore,  a  steel  with  an  elastic  limit 
of  45,000  to  50,000  lbs/in2  is  desirable,  otherwise  the  elastic 
limit  of  the  steel  will  need  to  be  taken  into  account  in  deter- 
mining the  ultimate  strength,  and  in  estimating  the  real  factor 


190  WORKING  STRESSES.  [On.  V, 

of  safety  and  hence  the  working  stresses.  The  behavior  of 
mild-steel  in  columns  when  stressed  beyond  its  elastic  limit 
is  not  well  determined.  Tests  where  mild  and  hard  steels 
have  been  used  side  by  side  show  less  difference  in  ultimate 
strength  than  would  be  expected.  Supported  by  the  surround- 
ing concrete,  buckling  cannot  take  place  until  the  concrete  fails, 
hence  the  resistance  of  the  rods  in  compression  is  probably 
greater  than  in  an  ordinary  compression  test. 

139.  Use  of  Steel  at  Ordinary  Working  Stresses.—  From  the 
preceding  discussion  it  is  seen  that  so  long  as  the  stresses  in 
the  concrete  are  kept  within  ordinary  working  values  of  400  to 
500  lbs/in2  the  stress  in  the  steel  will  be  much  below  usual 
working  limits.  Thus  with  a  value  of  400  for  fc  and  n  =  15,  f~8  is 
only  6000  lbs/in2,  or  less  than  one-half  its  safe  value.  Under 
these  conditions  it  is  a  question  wliether  it  may  not  be  more 
advantageous  to  use  a  higher  working  stress  in  the  steel 
and  place  little  or  no  dependence  on  the  concrete  for  carrying 
direct  stress.  The  relation  of  the  necessary  quantities  of  steel 
and  of  working  stresses  for  such  a  case  as  compared  to  the 
usual  reinforced  column  will  be  determined. 

If    A  =  total  area  of  reinforced  concrete  column; 
pA  —  area  of  steel  in  the  reinforced  column  ; 
Aa'  =area  of  steel  in  a  steel  column  using  customary  work- 

ing stresses; 

/c  =  the  working  stress  in  the  concrete; 
/8=the  usual  working  stress  in  the  steel  for  an  all-steel 

column  ; 
then  for  equal  strength 


(1) 


Assuming  the  areas  of  the  steel  equal  in  the  two  cases  (p  A  = 
and  solving  for  p  we  get 


§  140.]  COLUMNS.  191 

This  is  the  percentage  of  steel  which,  if  used  in  the  reinforced 
column,  will  give  the  same  total  section  of  steel  as  will  be 
required  in  an  all-steel  column  under  the  working  stress  fa. 
If,  for  example,  /<,=400,  ft  =  15,  and  /8  =  16,000,  we  have 
p = 400  +  (16,000  - 400  X 14)  =  3.8% .  This  calculation  indicates 
that  columns  reinforced  with  large  amounts  of  steel  are  not 
likely  to  compare  favorably  in  cost  with  the  all-steel  column, 
although  the  elements  of  pound  cost  of  steel  and  of  fireproofing 
must,  of  course,  be  considered. 

The  question  now  arises  as  to  when  a  combination  steel- 
concrete  column  should  be  calculated  as  a  reinforced  column  and 
therefore  with  reference  to  safe  stress  on  the  concrete  arid  when 
it  may  be  calculated  as  an  all-steel  column  merely  surrounded 
by  concrete.  Evidently  where  the  steel  is  used  only  in  small 
sections  and  depends  mainly  upon  the  concrete  for  rigidity  the 
column  must  be  calculated  with  reference  to  the  safe  concrete 
stress,  but  where  the  steel  is  in  a  form  to  be  able  to  act  as  a 
column  independently  of  the  concrete,  then  it  would  be  proper 
to  calculate  the  strength  in  either  way.  As  to  how  much 
should  be  allowed  for  the  concrete  when  the  steel  in  such  a 
column  is  stressed  up  to  16,000-18,000  lbs/in2  is  uncertain. 
Ordinarily  nothing  is  allowed,  but  undoubtedly  the  ultimate 
strength  of  such  columns  is  very  considerably  increased  by  the 
surrounding  body  of  concrete.  It  would  seem  that  a  moderate 
amount  of  hooping  would  in  this  case  be  very  advantageous. 
It  would  render  the  concrete  "  tough  "  and  reliable  under  the 
relatively  large  deformations  corresponding  to  the  working 
stress  in  the  steel.  This  would  enable  it  to  be  depended  upon 
for  a  certain  amount  of  resistance.  Tests  on  this  form  of  col- 
umn are  much  needed. 

DURABILITY   OF  REINFORCED   CONCRETE. 

140.  The  Protection  of  Steel  from  Corrosion. — A  continuous 
coating  of  Portland  cement  has  been  found  by  experience  to 
be  a  practically  perfect  protection  of  steel  against  corrosion. 
The  rusting  of  iron  requires  the  presence  of  moisture  and  carbon 


192  WORKING  STRESSES.  [Cn.  V. 

dioxide.  Portland  cement  not  only  forms  a  coating  which  ex- 
cludes the  moisture  and  C02,  but  in  hardening  it  absorbs  C02, 
tending  to  remove  any  of  this  gas  which  may  be  present.  In 
practice  the  protective  nature  of  Portland-cement  concrete  has 
long  been  known,  and  its  use  as  a  paint  was  adopted  by  the 
Boston  Subway  Engineers  after  careful  investigation. 

While  an  unbroken  coating  of  cement  offers  what  appears 
to  be  a  perfect  protection,  the  value  of  a  concrete  as  actually 
deposited  may  be  very  much  less.  A  series  of  experiments 
made  by  Professor  Charles  L.  Norton  gives  valuable  information 
on  this  subject.  In  one  series,  small  specimens  of  steel  6"  long 
were  embedded  in  blocks  3"X3"x8"  in  size  of  various  mix- 
tures of  cement,  sand,  and  stone  or  cinders.  The  blocks  were 
then  exposed  for  three  weeks  to  various  corrosive  atmospheres 
consisting  of  steam,  air,  and  C02.  The  results  were  as  follows: 
The  neat  cement  furnished  perfect  protection.  The  specimens 
embedded  in  mortars  and  concretes  showed  spots  of  rust  at 
voids  or  adjacent  to  a  badly  rusted  cinder.  He  concludes  that 
concrete  to  be  an  effective  protection  should  be  mixed  quite 
wet  so  as  to  furnish  a  thin  coating  on  the  metal,  and  must  be 
free  from  voids  and  cracks.  He  finds  that  dense  cinder  concrete 
mixed  wet  is  as  effective  as  stone  concrete. 

In  a  second  series  of  experiments  on  steel  already  rusted, 
from  a  slight  stain  to  a  deep  scale,  the  following  results  were 
obtabed :  The  concrete  was  1 : 2  J  :  5  (stone)  and  1:3:6 
(cinders).  After  one  to  three  months  in  corroders  and  one  to 
nine  months  in  damp  air  no  specimen  showed  any  change 
except  where  the  concrete  was  poorly  applied.  Some  of  the 
concrete  was  purposely  made  very  dry  and  the  rods  were  not 
well  covered.  These  specimens  were  seriously  corroded.  Un- 
protected steel  specimens  subjected  to  the  same  treatment  were 
almost  entirely  corroded.  While  the  experiments  of  Professor 
Norton  provided  for  a  covering  of  1J  inches,  there  is  no  reason 
to  suppose  that  a 'much  thinner  covering,  if  intact,  will  not 
furnish  as  good  protection. 

Many  cases  have  been  cited  of  steel  removed  from  concrete 


§  141.]  FIREPROOFING  EFFECT  OF  CONCRETE.  193 

after  the  lapse  of  20  years  or  more  and  found  to  be  in  perfect 
condition.  A  test  by  Mr.  H.  C.  Turner,*  in  which  steel  bars 
embedded  to  a  depth  of  3  inches  in  blocks  of  1:2:4  and  1:3:5 
concrete  and  exposed  to  sea-water  and  air  for  nine  months 
showed  perfect  preservation. 

141.  Fireproofing  Effect  of  Concrete. — Severe  fire  tests 
show  that  when  concrete  is  subjected  to  red-hot  temperatures 
(about  1700°)  for  three  or  four  hours  and  then  is  quenched  by 
hose  streams,  it  is  likely  to  show  pitting  but  that  it  will  still 
offer  a  sufficient  protection  to  the  steel. | 

A  reinforced-concrete  building  at  Bayonne,  N.  J.,  was  sub- 
jected to  a  very  hot  fire  in  the  burning  up  of  its  contents  but 
with  no  injury  to  the  building.  { 

In  the  Baltimore  fire  of  1904  the  value  of  concrete  as  a 
fireproofing  material,  and  of  reinforced-concrete  construction, 
was  fully  demonstrated.  Professor  C.  L.  Norton  of  the  Insur- 
ance Engineering  Experiment  Station,  after  a  careful  study  of 
the  damage  done  by  the  fire,  states  as  follows :  § 

"Where  concrete  floor  arches  and  concrete-steel  construc- 
tion received  the  full  force  of  the  fire  it  appears  to  have  stood 
well,  distinctly  better  than  the  terra-cotta."  The  reason  for 
this  he  considers  to  be  the  fact  that  terra-cotta  expands  about 
twice  as  much  as  steel,  but  that  concrete  expands  about  the 
same.  Little  difference  was  observed  between  stone  and 
cinder  concrete.  High  temperatures  long  continued  dehydrate 
and  soften  concrete,  but  this  process  in  itself  gives  off  water  and 
absorbs  the  heat,  thus  protecting  the  interior.  The  layer  of 
changed  material  is  then  a  better  non-conductor  than  before, 
so  the  process  goes  on  very  slowly.  Captain  J.  S.  Sewell,  report- 
ing to  the  Chief  of  Engineers  ||  on  the  Baltimore  fire,  states 
that,  with  reference  to  concrete  construction  subjected  to  very 

*  Eng.  News,  Aug.,  1904,  p.  153. 

t  See  tests  by  Professor  Ira  W.  Winslow  in  Eng.  Record,  Nov.  26,  1904, 
p.  634,  and  by  Professor  F.  P.  McKibben  in  Eng.  News,  Nov.  21, 1901,  p.  378. 
J  Eng.  Record,  April  12,  1902,  p.  341. 
§  Eng.  News,  June  2,  1904,  p.  524. 
||  Eng.  News,  March  24,  1904. 


194  WORKING  STRESSES.  [On.  V. 

high  heats:  " Exposed  corners  of  columns  and  girders  were 
cracked  and  spalled,  showing  a  tendency  to  round  off  to  a  curve 
of  about  3  in.  radius.  Where  the  heat  was  most  intense  the  con- 
crete, was  calcined  to  a  depth  of  J"-|",  but  showed  no  tendency 
to  spall,  except  at  exposed  corners.  On  wide,  flat  surfaces 
the  calcined  material  was  not  more  than  J-in.  thick  and  showed 
no  disposition  to  come  off.  The  terra-cotta  fireproofing  showed 
up  much  poorer."  In  his  general  conclusions  he  considers  it 
at  least  as  desirable  as  steel  work  protected  by  the  best  com- 
mercial hollow  tiles,  and  preferable  to  tile  for  floor  slabs  and 
fire-proof  covering.  While  satisfactory  protection  of  the  steel 
can  thus  doubtless  be  secured  the  effect  of  fire  upon  the  con- 
crete itself,  and  its  usefulness  after  more  or  less  calcination,  is 
a  question  of  the  utmost  importance  and  one  on  which  much 
more  information  is  needed. 

The  necessary  thickness  of  concrete  to  furnish  adequate 
fire  protection  depends  somewhat  upon  the  character  and  im- 
portance of  the  member.  Such  members  as  main  girders,  where 
a  failure  would  involve  a  considerable  portion  of  the  building 
and  where  the  steel  is  concentrated  in  a  few  rods,  should  be 
more  thoroughly  protected  than  floor  slabs  of  small  span,  where 
&  few  local  failures  would  be  of  no  importance,  and  where  addi- 
tional covering  would  add  largely  to  the  expense.  Results  of 
fire  tests  and  experience  in  conflagrations  indicate  that  2"-2J" 
will  offer  practically  complete  protection,  and  that  a  minimum 
of  J"-f"  for  floor  slabs  will  usually  be  sufficient.  Large  flat 
surfaces,  such  as  floor  slabs,  are  less  exposed  than  the  corners 
of  projecting  forms  like  beams  and  columns.  In  a  report  of  a 
committee  of  members  of  the  American  Society  of  Civil  Engi- 
neers on  the  effects  of  fire  in  the  San  Francisco  conflagration, 
similar  conclusions  were  reached  as  to  the  value  of  concrete  as 
a  fire-proofing  material.  It  was  also  found  far  preferable  to 
tile  for  floors.  With  respect  to  the  injury  to  the  concrete  itself 
the  committee  was  of  the  opinion  that  it  was  sufficient  in  many 
cases  to  require  reconstruction.* 

*  Proc.  Am.  Soc.  C.  E.,  March,  1907. 


$  142.]        SHRINKAGE  AND  TEMPERATURE   STRESSES.  195 

142.  Reinforcing  Against  Shrinkage  and  Temperature 
Stresses. — Where  a  reinforced  structure  is  unrestrained  by 
outside  forces  the  only  stresses  arising  from  shrinkage  and 
temperature  changes  are  those  due  to  the  mutual  action  of  steel 
and  concrete.  As  the  two  materials  have  nearly  equal  rates  of 
expansion  temperature  changes  will  cause  very  little  stress. 
Shrinkage  in  hardening  will  cause  more  important  stresses,  as 
shown  in  Art.  43,  but  still  not  unduly  large  unless  the  steel 
ratio  is  very  high. 

When  the  structure  is  restrained  by  outside  forces  so  that 
it  is  not  free  to  contract  or  expand,  as  in  the  case  of  a  long  wall, 
then  the  resulting  stresses  are  likely  to  be  high.  When  not 
reinforced,  concrete  will,  under  such  circumstances,  crack  at 
intervals,  its  maximum  deformation  under  stress  not  being 
equal  to  its  maximum  temperature  deformations.  If  it  be 
assumed  that  concrete  when  reinforced  will  not  stretch  more 
than  plain  concrete,  as  seems  probable  (Art.  42),  then  no  amount 
of  reinforcement  can  entirely  prevent  contraction  cracks.  The 
reinforcement  can,  however,  force  such  cracks  to  take  place 
as  they  do  in  a  beam — at  such  frequent  intervals  that  the 
requisite  deformation  takes  place  without  any  one  crack  be- 
coming large.  Laboratory  tests  on  beams  would  indicate  that  if 
steel  is  used  in  sufficient  quantities  the  cracks  may  easily  remain 
quite  invisible  and  be  of  no  consequence  from  any  practical 
standpoint.  Thus  if  the  coefficient  of  expansion  be  .000006 
a  change  of  temperature  of  50°  causes  a  change  of  length 
(if  free)  of  .0003  part.  A  deformation  of  this  amount  in  a 
beam  (corresponding  to  a  steel  stress  of  9000  lbs/in2)  would 
not  cause  cracks  easily  detected.  The  prevention  of  large 
cracks  by  means  of  reinforcement  is  then  a  matter  of  using 
•sufficient  steel  to  force  the  concrete  to  crack  at  small  intervals. 
No  one  crack  will  open  up  far  until  the  steel  is  stressed  beyond 
its  elastic  limit,  hence  we  may  say  approximately  that  the 
amount  of  steel  used  must  be  such  that  the  concrete  will  crack 
elsewhere  before  the  steel  is  stressed  beyond  its  elastic  limit. 
A  larger  amount  of  steel  will  serve  to  keep  the  cracks  smaller. 


196  WORKING  STRESSES  [Cn.  V. 

In  calculating  the  requiste  amount  of  steel  the  temperature 
stress  in  the  steel  itself  must  be  considered.  This  will  add  to 
its  skrinkage  stress,  so  that  its  total  stress  will  equal  its  tempera- 
ture stress  plus  the  stress  necessary  to  crack  the  concrete.  If, 
for  example,  the  assumed  drop  in  temperature  be  50°  the  tem- 
perature stress  in  the  steel  =  50  X  .0000065X30,000,000  =  9750 
lbs/in2.  If  the  tensile  strength  of  the  concrete  be  200  lbs/in2 
and  the  assumed  allowed  stress  (elastic  limit)  in  the  steel  be 
40,000  lbs/in2,  then  the  stress  available  =40,000 -9750  =  30,250 

200 
lbs/in2,  and    the    required    percentage   of   steel  =  p  =  Q         •  = 

oU,^OU 

.0066.    If  the  elastic  limit  be  60,000  lbs/in2  the  steel  ratio  = 

200 
P  =  6Q  QQQ  _Q75Q =  -004.     For    the    purposes    here    considered 

obviously  a  high  elastic-limit  steel  is  desirable,  and  in  order 
to  distribute  the  deformation  as  much  as  possible  a  mechanical 
bond  is  advantageous. 


CHAPTER  VI. 

FORMULAS,  DIAGRAMS,  AND  TABLES. 

103.  Rectangular  Beams;   Linear  Variation  of  Stress, 


jd 


FIG.  64. 
Notation. 

}8  =unit  stress  in  steel; 

}c=  "       "     "  concrete; 
Es= modulus  of  elasticity  of  steel; 
Ec=      "        "        "        "  concrete; 

n=Eg/Ec', 

T= total  tension; 

C=    "     compression; 

Ms  =  moment  of  resistance  relative  to  the  steel; 
Mc=      "        "        "  "       "     "  concrete; 

Af = bending  moment  or  moment  of  resistance  in  general; 
A=  steel  area; 

b=  breadth  of  beam; 

d=net  depth  of  beam; 

k  =  ratio  of  depth  of  neutral  axis  to  depth  d] 

j= ratio  of  lever- arm  of  resisting  couple  to  depth  d] 

197 


198  FORMULAS,  DIAGRAMS,  AND   TABLES.  [Cn.  VI. 

p  =  steel  ratio  =  A/bd  ; 
Es=f8pj  =  "  coefficient  of  strength"  relative  to  steel; 

"         "       "  '••       "  concrete. 


Formulas. 

Position  of  neutral  axis, 


k=V2pn-\-(pn)2—jm  ......     (1) 

Arm  of  resisting  couple, 

/  =  1-P.       .......     (2) 

Moment  of  resistance, 

?=R8'b<P,       .....     (3) 

d*  =  Re-bd?  ......     (4) 

Approximately, 


(3') 
(4') 


Fibre  stresses, 

T 


''    A        A     ' 


2C_ 

Jc~bkd~     bkd 
Steel  ratio, 

......     (7) 


Cross-section  of  beam  for  given  bending  moment  M, 

•*-£-!  ...... 


§  144.]  RECTANGULAR  BEAMS.  199 

Diagrams. — Plates  I-IV,  pp.  213-216,  are  diagrams  of  values 
of  k  and  /  for  various  values  of  p-,  and  values  of  R8  and  Rc 
(called  simply  R)  for  various  values  of  p  and  of  fs  and  fc.  The 
value  of  n  is  taken  at  10,  12,  15,  and  18  respectively. 

The  use  of  the  diagrams  in  rinding  moments  of  resistance 
(Eqs.  (3)  and  (4))  and  in  determining  cross-sections  (Eqs.  (8) 
and  (9))  is  obvious.  The  proper  steel  ratio,  p,  to  use  for 
given  values  of  f8  and  fc  (Eq.  (7))  is  determined  from  the 
intersection  of  the  curves  for  the  given  values  of  fs  and  fc. 
Finally,  the  actual  fibre  stress,  /8  or  fct  resulting  from  a  given 
M,  p,  and  bd2  will  be  found  by  first  calculating  M/bd2  from 
the  given  values.  Call  this  R.  Then  with  this  value  of  R  and 
the  given  value  of  p  enter  the  diagram  and  find  the  corre- 
sponding values  of  /,  and  /c. 

ILLUSTRATIVE  EXAMPLES.  —  1.  Moment  of  Resistance.  —  Given  the 
following:  6  =  12",  d=20",  /«  =  14,000,  /c-600,  and  p=0.8%;  find  M9 
and  Mc.  Assume  n  =  15.  Solution.  From  Plate  III,  p.  215,  we  find  for 
p=Q.8%  and  /«  =  14,000,  #8=96;  and  for  /c=600,  #c  =  100.  Hence 
M8  =966^  =460,800  in-lbs.,  and  Mc  =  l 006*^=480, 000  in-lbs. 

2.  Fibre  Stresses.— Given  6  =  12",  d=20",  p=0.8%,  and  M  =450,000 
in-lbs.,  to  find  /«  and  fc.    Solution.  Use  Eqs.  (5)  and  (6)  directly;    or, 
find  M/bd?  and  use  the  diagrams.    Thus  M/bd*=  450,000/4800  =93.75. 
Then  from  Plate  III,  with  #=93.75  and  p=0.8%  we  find  /«=  about 
13,500  and  fc=  about  560  lbs/in2. 

3.  Cross-section  of  Beam  and  Steel  Ratio. — Given  M  =500,000  in-lbs., 
fs  =  12,000,  /c=500,  to  find  btf.    Solution.  From  Plate  III  we  find  at 
the  intersection  of  the  curves  for  fs  =  12,000  and  /c=500,  a  value  of 
R  of  84.     Hence  6cP=  500, 000/84  =5950.    The  required  amount  of  steel 
is  also  found  from  the  diagram  to  be  0.8%. 

144.  Rectangular  Beams ;  Parabolic  Variation  of  Stress ; 
for  Ultimate  Loads. 

Notation. — 

As  in  Art.  143,  but  here  Rc  =  $fJcj. 

Formulas. 

Position  of  neutral  axis, 

(10) 


200  FORMULAS,  DIAGRAMS,  AND  TABLES.  [Cn.  VI, 

Arm  of  resisting  couple, 


Moment  of  resistance, 

Ms=fspj-bd2=R8-bd2, (12) 

Mc  =  $fcki-bd2=RC'bd2 (13) 

Approximately, 

M8=fsA.Q.8d,       (120 

Mc=fc>Q.28bd2 (13') 

A  /. 


FIG.  65. 


Fibre  stresses, 


Steel  ratio, 


IS  A  \  > 

jL\.  ^i 

fc  =  bkd^     bkd    '  *, 
1 


(14) 
(15) 
(16) 


Cross-section  of  beam  for  given  bending  moment  M, 


(17) 


(18) 


§  145.] 


T-BEAMS. 


201 


Diagrdms. — Plate  V,  p.  217,  is  a  diagram  of  values  of  k  and 
j  for  various  values  of  p\  and  values  of  Rs  and  Rc  for  various 
values  of  p,  fs,  and  /c.  The  full  lines  are  drawn  for  ft  =  15;  the 
dotted  lines  for  n=12.  The  fibre  stresses  are  here  assumed  as 
representing  ultimate  strengths,  and  the  diagram  is  supposed 
to  give  results  pertaining  to  ultimate  strength.  To  use  it 
for  purposes  of  designing,  the  given  loads  or  moments  should 
be  multiplied  by  the  selected  factor  of  safety,  or  the  value 
of  R  obtained  from  the  diagram  divided  by  such  factor  of  safety. 

145.  T-Beams ;  Linear  Variation  of  Stress. 

.  /c 


I 

1  k,        - 
1 

1 

| 

•  1 

»• 

*; 

<  

fr-» 

FIG.  66. 

Notation.     (In  addition  to  that  of  Art.  143.) 
6= width  of  flange; 
fr'  =  width  of  web; 
t= thickness  of  flange; 
c= depth  of  neutral  axis; 
x = depth  of  resultant  of  compressive  stress; 
d— £  =  arm  of  resisting  couple. 

Formulas. 

Case  I.  Neutral  axis  in  the  flange. 

Use  formulas  (l)-(9)  as  for  rectangular  beams; 
formula  (1)  for  k  will  determine  whether  the  case 
is  I  or  II. 


Approximately, 


M 


(19) 

(20) 


202  FORMULAS,  DIAGRAMS,  AND  TABLES.  [Cn.  VI. 

Case  II.  Neutral  axis  in   the  web;     compression  in  web 
neglected. 

Position  of  neutral  axis, 


Position  of  resultant  of  compressive  stress, 
3c-2t  t 


Steel  area, 


Moment  of  resistance, 

M9=f.A(d-x),       .......     (23) 


Approximately, 

M8=fsA(d-U),        .....     (23'} 


Approximately, 

'  .......  (25'> 


146.  Beams  Reinforced  for  Compression. 

Notation.     (In  addition  to  that  of  Art.  143.) 
Af  =  area  of  compressive  steel; 
p'  =  steel  ratio  of  compressive  steel; 
//=  unit  stress  in  "  " 

C"  =  total  stress  in  the  compressive  steel; 
d!  =  distance  from  compressive  face  to  the  plane  of  the  com- 
pressive steel; 
s=depth  to  resultant  compression,  C+C'. 


§  146.]  BEAMS  REINFORCED  FOR  COMPRESSION.  203 

Formulas. 

Position  of  neutral  axis, 


.  .     .     (26) 
Position  of  resultant  of  compressive  stress,  C+C", 


d'C'  I      d'\ 

k+~dc  r   2P'n(k+j) 

/ 


x=- £7~;  in  which  — = 


.   .     .     (27) 


Arm  of  resisting  couple, 
Moment  of  resistance, 

Fibre  stresses, 


M+jd 

A    ' 

k 


L    df\ 
n{  k—j]         ,  ,     ,. 

;,__\ it    f       M~d 

/«  7.  ic—   j       ]^J 


(28) 


(29) 

-d').    .    .    .     (30) 


(31) 

(32) 


204 


FORMULAS,  DIAGRAMS,  AND  TABLES. 


[On.  VI. 


Diagrams. — Values  of  k  and  j  are  given  in  Fig.  29,  p.  86, 
for  various  values  of  p  and  of  p'.  It  is  assumed  that  d'/d 
=  1/10  and  n  =  15.  Plate  VI,  p.  218,  gives  the  amount  of  com- 
pressive  steel  (values  of  p')  necessary  to  use  in  order  to  reduce 
the  compressive  fibre  stress,  /c,  any  given  percentage  below 
the  value  it  would  have  with  no  compressive  reinforcement. 
The  effect  of  this  compressive  steel  upon  the  value  of  the  ten- 
sile stress  in  the  steel  is  also  given  in  the  diagram  for  various 
values  of  p  and  p' . 

147.  Flexure  and  Direct  Stress.— There  are  two  cases: 
I.  Where   there   is   compression   on   the   entire   cross-section 

(Figs.  68  and  69); 
II.  Where  there  is  some  tension  on  the  cross-section  (Fig.  70). 


fc 


FIG.  68. 


FIG.  70. 


§  147.]  FLEXURE  AND  DIRECT  STRESS.  205 

Notation. — The  lower  side  of  the  beam  in  the  figures  on  the 
preceding  page  is    called  the  " tension  face". 

R  =  resultant  force  acting  on  the  section ; 

N  =  component  of  R  normal  to  section ; 
e  =  eccentric  distance  of  R,  e/h  =  eccentricity; 

M = bending  moment  =  Ne ; 

A'  =  area  of  steel  near  compressive  face ; 

pf=A'/bh; 

A  =  area  of  steel  near  tension  face ; 

p=A/bh] 

df  =  distance  of  compressive  steel  from  face; 

u= distance  from    compressive  face  to  centroid  of  trans- 
formed section; 

a = distance  from   steel  to  center   of  section  for  symmet- 
rical reinforcement; 

^  =  area  of  transformed  section; 

Ic=  moment   of  inertia  of  concrete  about  central  axis  of 
transformed  section; 

I8=  moment  of  inertia  of  steel  about  central  axis  of  trans- 
formed section; 

It=  moment  of  inertia  of  transformed  section; 

fc=  maximum  compressive  fibre  stress  in  concrete; 

//=  maximum  tensile  fibre  stress  in  concrete; 

fo  =  stress  in  steel  near  compressive  face; 

fs  =  stress  in  steel  near  tension  face ; 


Formulas. 
General. 


At=bh+n(A+A')  .......     (34) 

.....     .     .     .    .    .     (35) 


u= 


l  +  np+np' 


206  FORMULAS,  DIAGRAMS,  AND  TABLES.  [Cn.  VI. 


Case  I.  Compression  on  the  entire  cross-section. 
Fibre  stresses: 


(37) 


N     M(h-u) 

]c    =  ~       -  , 


If  //  is  negative,  then  the  case  is  Case  II. 
For  symmetrical  reinforcement  and  for  d'/d  = 


Case  II.  Tension  on  part  of  the  cross-section. 

If  the  tension  in  the  concrete  is  considered,  use  the  formu- 
las of  Case  I. 

For  symmetrical  reinforcement  and  for  d'/d  = 

• 


(43) 


If  the  tension  in  the  concrete  is   neglected. 
For  symmetrical  reinforcement  and  for  d'/d 


(46) 


kh-d' 


-    •    •    •    (47) 


§  148.]  SHEARING  AND  BOND   STRESS.  207 

Diagrams. — Values  of  l/k  for  Case  I,  Eqs.  (41)  and  (42),  and 
Case  II,  Eqs.  (43)  and  (44),  are  given  in  Fig.  33,  p.  95;  and 
values  of  k  for  Case  II,  Eqs.  (45)  and  (46),  are  gwen  in  Fig.  35, 
p.  98.  Plate  VII,  p.  219,  is  a  diagram  for  values  of  M/bh2f0 
for  Case  I,  Eq.  (41);  and  Plate  VIII  for  the  same  quantity 
for  Case  II,  Eq.  (45),  given  in  terms  of  the  eccentricity  e/h 
and  the  steel  ratio  p.  The  diagrams  are  constructed  for  n  =  15. 

ILLUSTRATIVE  EXAMPLES. — I.  An  arch  ring  is  24  in.  deep  and  is 
symmetrically  reinforced.  For  each  side  p=0.9%.  On  a  width  of 
12  in.  N  =75,000  Ibs.;  e=3  in.;  what  is  the  maximum  stress  /c? 
Solution.  The  eccentricity  =3/24  =  .125.  The  diagram  of  Plate  VII 
will  be  used,  and  the  case  is  Case  I.  This  diagram  gives  at  once 
MM2/C  =  .097.  We  also  have  M  =75,000x3=225,000  in-lbs.  Hence 
fc  =225,0007(12 X242 X .097)  =336  lbs/in2. 

2.  If,  in  Ex.  1,  the  eccentricity  be  6  in.,  find  the  maximum  compres- 
sive  stress  fc  and  the  maximum  tensile  stress  /</,  the  concrete  being  con- 
sidered as  carrying  tension  if  necessary.     Solution.  Use  Plate  VII.     The 
eccentricity  is  6/24  =  .25.     From  the  diagram  we  find  M/bh2fc  =  .I4l, 
whence  /c=572    lbs/in2.      From    Eq.  (10),  p.  94,  the  value  of  A;  =  .9. 
This  being  less  than  unity  there  will  be  tension  on  the  section.     From 
Eq.  (43)  the  tensile  concrete  stress  =//= 64  lbs/in2. 

3.  If  in  Ex.  2  the  tension  in  the  concrete  be  neglected,  find  fC)  /a',  and 
fs.    Solution.     Use  Plate  VIII.     e/7t  =  .25.     The  value  of  M/bh2fc  =  .U, 
whence  /c=576  lbs/in2.      The  compressive  stress  in  the  steel,  /«',  is 
always  less  than  nfc ;    in  this  case  it  is,  from  Eq.  (46),  equal  to  nfc  X 

(l—  — y)  =n/c  X  .88,  k  being  found  from  Fig.  35.     The  tensile  steel  stress, 
\       1 1/c/ 

f8,  is  less  than  the  compressive.     From  Eq.  (47)  it  is  found  to  be  276 
lbs/in2. 

148.  Shearing  and  Bond  Stress. 
Notation. 

V  =  total  vertical  shear  at  any  section; 
v  =  maximum  horizontal  or  vertical  shearing  stress  per 

unit  area; 

i/  =  average  shearing  stress  per  unit  area; 
U  =  bond  stress  per  unit  length  of  beam; 
b  and  d  =  dimensions  of  a  rectangular  beam; 
&'  =  width  of  web  of  T-beam ; 


208  FORMULAS,  DIAGRAMS,  AND  TABLES.  [Cn.  VI. 


d  =  net  depth  of  T-beam; 
t  =  flange  thickness  of  T-beam; 
jd  =  arm  of  resisting  couple  for  any  beam. 

Formulas. 

Rectangular  beams  : 

•'-SP-. 


p-    ......    .....    (50) 

Approximately, 


tf  =  y^ (500 

T-beams : 

V 


u=  Jd 

Approximately, 


,     ....  |   ^-arq 

149.  Columns. 


^4.  =  total  cross-section; 
Ac  =  cross-section  of   concrete; 
A8=         "  "    longitudinal  steel; 


(52'} 


§  149.]  STRESSES  IN  CIRCULAR   PLATES.  209 

P=AS/A- 

P  =  strength  of  plain  concrete  column; 
Pr  =      "          "  reinforced  column; 

/c  =  unit  stress  in  concrete; 

fs  =    "       "       "    steel  (not  exceeding  its  elastic  limit)  ; 
I  ei  =  elastic-limit  strength  of  steel  ; 

/  =  average  unit  stress  for  entire  cross-section; 
//=  steel  ratio  of  the  hoops  of  hooped  columns. 

Formulas. 

For  short  columns;  ratio  of  length  to  least  width  not  ex- 
ceeding 20: 

f,  =  nfe,    ..........     (54) 

P'-fcAc+f.A.,      .......     (55) 

Pf=feA[l  +  (n-l)p]t      .....     (56) 

(57) 


If  nfc  is  greater  than  the  elastic-limit  strength  of  the 
steel,  then 

P'=fcAc+felAs  .......     (58) 

Considered  formula  for  hooped  columns: 

P'=fcAc+Up+2Ap')A.  .....    (59) 

For  long  columns: 


2' 


10,000  \r 

Diagrams.  —  Plate  IX  is  a  diagram  of  the  function 
l  +  (n-l)p  (=///c)  of  Eqs.  (56)  and  (57)  for  various  values 
of  p  and  values  of  n  equal  to  10,  12,  15,  20,  and  25.  The  aver- 
age working  stress,  /,  for  any  column  is  then  found  by  mul- 
tiplying the  corresponding  ordinate  from  this  diagram  by  the 
selected  working  stress  fc. 


210  DIAGRAMS,  FORMULAS,  AND  TABLES.  [On.  VI. 

150.  Stresses  in  Circular  Plates. — The  exact  determina- 
tion of  stresses  in  floor  systems,  such  as  the  "mushroom"  sys- 
tem described  in  Art.  168,  and  in  the  ordinary  foundation- 
plate  supporting  a  single  column,  involves  very  complex 
analytical  processes.  As  an  aid  in  estimating  the  stresses  in 
such  cases,  Plates  X  arid  XI  have  been  prepared.  They  give 
the  bending  moments  in  circular  plates  supported  rigidly  over 
any  given  area  at  the  center.  Plate  X  gives  the  moments  for 
the  case  of  a  uniformly  distributed  load  on  the  entire  area, 
and  Plate  XI  the  moments  for  a  load  uniformly  distributed 
along  the  periphery.  In  each  case  the  full  lines  give  the 
coefficients  for  the  radial  bending  moments,  and  the  dotted 
lines  those  for  the  circumferential  bending  moments.  The 
curves  are  drawn  for  five  different  ratios  of  r±  to  TO,  or  radius 
of  plate  to  radius  of  fixed  support.  For  other  ratios  interpo- 
lations may  be  made. 

The  calculations  for  the  diagrams  are  based  upon  the 
analysis  presented  by  Prof.  H.  T.  Eddy*  for  homogeneous 
plates.  The  value  of  Poission's  ratio  assumed  in  the  numerical 
substitutions  has  been  0.1,  as  approximately  determined  in 
recent  experiments  by  Prof.  A.  N.  Talbot. 

Example. — A  circular  plate  10  ft.  in  diameter  is  rigidly  supported 
by  a  column  24  in.  in  diameter.  It  supports  a  load  of  150  lbs/ft2  over 
the  area  and  a  load  of  500  lbs/ft  along  its  outer  circumference.  Re- 
quired, the  radial  and  circumferential  bending  moments. 

Solution.  The  ratio  of  r1:rQ  =  120:  24  =  5.  (The  upper  diagram  of 
Plate  XI  may  be  used  in  finding  this  ratio.)  In  Plate  X  we  then  obtain 
the  coefficients  Ql  and  Q2  for  any  desired  point  in  the  plate,  using  the 
curves  corresponding  to  rl  -f-r0  =  5.  The  value  of  Qj  (ordinate  to  dotted 
curve)  is  seen  to  be  a  maximum  at  a  distance  from  the  center  equal  to 
about  1.7r  ;  its  value  is  about  4.7.  Hence  the  maximum  circumferen- 
tial moment  due  to  the  load  of  150  lbs/ft  is  4.7xl50xl2=705  ft-lbs 
per  foot  width  of  section.  The  value  of  Q2  (ordinate  to  full  curve)  is  a 
maximum  at  the  edge  of  the  support  and  has  a  value  of  16.  The  radial 
bending  moment  is  therefore  equal  to  16 x  150 X  I2  =  2400  ft-lbs  per  foot 

*Year  Book,  Engrs.  Soc.,  Univ.  of  Minn.,  1899. 


§  152.]  TABLES.  211 

width  of  section.    The  radial  moment  rapidly  falls  off  with  increased 
distance  from  the  support. 

The  moments  due  to  the  peripheral  load  of  500  Ibs/ft  are  found  from 
Plate  XI  to  be  respectively  M,  =  3. 1x500x1  =  1550  ft.-lbs.,  and 
M 2 = 9.6  X  500  x  1  =  4800  ft.-lbs. 

151.  Coefficients  and  Working  Stresses. — The  following  is 
a  resume  of  the  coefficients    and  working  stresses  suggested 
in  the  discussion  of  Chapter  V.     They  may  be  considered  as 
applicable   to  ordinary  conditions  on  the  basis  of  equivalent 
dead-load  stresses  and  with  concrete  of  1:2:4  to   1:2J:5  com- 
position. 

Beams. 

Working  Stress. 

Concrete  in  compression 500-600  -       lbs/in2 

Concrete  in  shear,  average  stress : 

a.  Without  shear  reinforcement.  30-40 

b.  With  shear  reinforcement.  .  .  50-80 

Bond  stress: 

a.  Smooth  rods 60-75  " 

•6.  Deformed  rods 100-175 

Steel  in  tension 12,000-15,000        " 

Value  of  n=Es/Ec 12-15 

Columns. 

Concrete  in  compression 300-400 

Value  of  n  =  Es/Ec. 15-20 

152.  Tables.— Areas   of   Steel   Rods— Table   No.  19   gives 
sectional  areas  and  weights  per  foot  of  round  and  square  rods 
of  various  sizes,  and  the  total  area  per  foot  of  width  of  slab 
when  the  rods  are  spaced  various  distances  apart. 

Materials  Required  for  One  Cubic  Yard  of  Concrete. — Table 
No.  20  gives  the  quantities  of  material  required  for  one  cubic 
yard  of  concrete  of  various  proportions.  The  table  is  based 


212  DIAGRAMS,  FORMULAS,  AND  TABLES.  [Cn.  VI. 

on  Thatcher's  Tables.*    As  conditions  vary  greatly,  these  tables 
should  be%  used  only  for  approximate  values. 

Soje  Loads  for  Floors. — Table  No.  21  gives  span  lengths  for 
floor- slabs  for  various  live  loads  per  square  foot,  and  for  various 
values  of  working  stresses  fs  and  fc.  The  tables  have  been 
calculated  for  beams  supported  at  the  ends,  the  bending  moment 
being  \wl2.  The  value  of  n  has  been  taken  at  15.  For  con- 
tinuous slabs  10  wZ2  is  commonly  taken  as  the  bending  moment. 
The  permissible  span  length  on  this  basis  will  be  12%  greater 
than  the  tabular  values.  Where  the  span  length  is  given,  to 
find  necessary  thickness  of  slab  based  on  yV^2>  take  90%  of 
the  given  span  length  and  look  up  this  value  in  the  table. 
The  table  also  gives  the  amount  of  steel  required  per  foot 
of  slab,  so  that  by  reference  to  Table  No.  18  a  suitable  size 
and  spacing  can  readily  be  determined.  The  moment  of 
resistance  of  a  beam  one  foot  wide  is  also  given  for  general 
use. 

*  Johnson's  Materials  of  Construction,  p.  610a. 


§  152.] 


TABLES. 

n=io 


213 


05  10  15 

Percentage  Keinforc  ment 


PLATE  I. 


214 


DIAGRAMS,  FORMULAS,  AND  TABLES.  [Cn.  VI. 


i.o 


05 


Pbrce 


1)0 
ntage  Rein 


merit 


Ue; 


of 


0.8 


0.7 


0.5 


160 


150 
140 


120 


110 
100 
,00 


70 


7 


10 


05  1JO 

Percentage  Reinforcement 


PLATE  II. 


§  152.] 


PLATE  III. 


216 


DIAGRAMS,  FORMULAS,  AND  TABLES. 


[CH.VI, 


Percentage  Reinforcement 


PLATE  IV. 


§  152.]  TABLES. 

Full  lines  for  n=  15 ;  dotted  lines  for  n 


217 


=  12. 


218 


DIAGRAMS,  FORMULAS,  AND  TABLES.  [On.  VI. 


Percentage  of  compressive  steel 
PLATE  VI. — Compressive  Reinforcement  of  Beams. 


§  152.] 


TABLES. 


219 


.02 


.08      ,10      .12       .14      .16      .18      .20      .22 


18 

17 
16 
15 
14 
13 
J3 
11 
.10 
05) 
08 
.07 
.06 
.05 
.04 
.03 


dS 


All  Values  of  M  I  &/r/c  are  based 
on  n  =  15  and  dl '.  h=  l/\Q 


X 


' 


>z 


I 


.01 


.02       .04        .06       .06       .10       .12       .14       .16       .18       .20       .22       .24 
Values  of  Eccentricity,  e-^-h 

PLATE  VII. — Flexure  and  Direct  Stress. 


220 


DIAGRAMS,  FORMULAS,  AND  TABLES. 


[Cn.  VI. 


.6  .8  1.0  1.2  1.4  1.6 

Values  of  Eccentricity,  e-i-h 

PLATE  VIII.— Flexure  and  Direct  Stress. 


1.8 


§  152.] 


TABLES. 


221 


Percentage  Reinforcement 


1.70 

/ 

i.ro 

1.60 
1.50 
1.40 
1.30 
1.20 
1.10 

1.0 

0 

/ 

/ 

1 

/ 

/ 

/ 

1.50 
1.40 
i  an 

/ 

/ 

/ 

J 

/ 

/ 

/ 

^x 

^ 

/ 

/ 

/ 

/ 

0 

• 

/ 

p 

v/ 

x 

X 

/ 

^x 

^ 

X 

s^- 

/ 

X 

x 

*4 

/ 

/ 

x 

n 

/ 

x* 

^ 

/ 

^ 

ce 

/ 

/ 

5 

x^ 

^x 

/ 

/ 

^ 

X 

x 

X 

1.90 
1  10 

x 

/ 

/ 

x' 

*\ 

1.^. 

x 

^ 

/ 

( 

X 

x* 

^x 

/ 

^ 

? 

X 

? 

x 

*^ 

\ 

x' 

x 

X1 

/ 

^ 

/ 

x 

x 

1 

/ 

, 

7 

/ 

/ 

X 

X 

^-* 

x" 

f^*^ 

/ 

/ 

f 

x 

^ 

X 

.X 

X" 

/ 

/ 

r 

., 

X 

x 

^ 

-^ 

? 

/ 

x 

x 

x' 

? 

/ 

/ 

, 

x 

x 

r^ 

^> 

x' 

/ 

/ 

x 

s 

^x 

•^ 

^ 

X" 

'•°o 

/ 

/ 

X 

x 

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^ 

-^ 

/ 

2 

x 

- 

X 

X 

•^ 

- 

4 

// 

x^ 

x^ 

X 

££; 

x** 

A 

^ 

.0                    0.5                    1.0                    1.5                    2.0                    2*5                    3, 

Percentage  Reinforcement 
PLATE  IX. — Working  Stresses  in  Columns. 


222 


DIAGRAMS,  FORMULAS,  AND  TABLES.  [Cn.  VI 


w 

ri~  ru| 

i  i  i  i  i  i 

H-Q,rt 

LLJLJ 

thof 
fiber- 

thof 

-.-ill 

e. 
3  in 

35 

r^ww^^  mm  wwmw'wm 

n^ 

MI  is  bending  moment,  per  unit  wid 
section,  causing  circumferential 
stress  at  any  distance  r. 
M2  is  bending  moment,  per  unit  wid 
section,  causing  radial  fiber-stre 
any  distance  7*. 
5    is  top  load  per  unit  area. 
Qj  is  ordinate  to  proper  dotted  curv 
Q2  is  ordinate  to  proper  solid  curve. 
If  q  is  expressed  in  Ibs.  per  sq.  ft 
and  jy  in  ft.,  then  M,and  M0will  b 
f  t.-lbs.  per  ft. 

1 

Cl      \ 

.25 

a 

1 

o 

1 

15 
10 
5 

I 

\ 

\ 

B 

\ 

\ 

\ 

\ 

\ 

\ 

s|      \ 

\ 

\ 

\ 

1  \ 

\ 

G    ! 

\- 

V, 

\/ 

\ 

c 

x^ 

\ 

\ 

V 

^^ 

10 

i\ 

\ 

\ 

x 

4, 

4 

>  \ 

\ 

\ 

"^v 

•^ 

y  f 

\ 

v^ 

>x 

_JA/ 

\ 

\ 

J4 

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A 

\ 

\ 

\ 

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,^_ 

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;  —  A 

v"^ 

\     \ 

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^^ 

r\^~^ 

3 

^    / 

Sy 

\ 

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V 

~-~~., 

-  i 

B 

—  t 

Av  ^, 

-X- 

> 

NV                    >x. 

""N; 

v~  —  ~ 

5 

f/\ 

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IS 

,  •** 

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^^ 

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1                    3                    3                    4                    5                    6                    7 

Values  of  r  T-  r0 
PLATE  X. — Stresses  in  Circular  Slabs. 


§  152.] 


TABLES. 


223 


ZT 


r  in  in. 


U. 


V7 


f 


I0are  explained  on  Plate  X. 
P    is  load  at  periphery  per  unit  length. 
P!    is  ordinate  to  proper  dotted  curve. 
{3    is  ordinate  to  proper  solid  curve. 

If  p  is  expressed  in  Ibs.  per  ft. 

and  r0  in  ft.,  then  iv^and  M2will  be  in. 

ft.-lbs.  per  ft.. 


s 


\ 


.X 


3  4 

Values  ofr-±-  rn 


PLATE  XI. — Stresses  in  Circular  Slabs. 


224 


DIAGRAMS,  FORMULAS,  AND  TABLES.  [Cn.  VI. 

^  CO' 

^ 


05 


**  "-* 

PH      Q 

il* 
J-Si 

32          «»     O 
5     °9     K 

H    W 
O 


$\ 


00 


(N 


•ft?' 

.Sfa 


111 

O  <B  « 

o£- 


T-i    T-H    rt    TH    (M 


1-H   1-H   1-H  1-1   (N   (N 


TH  i—  i  i—  i  i—  i  (N  (M  CO 


t^T-HCDCOTHr-ti—  lI>OOTfHO 


15- 


o"^ 


Mwwc^weococo^^ 


§  152..] 


TABLES. 


225 


"8 

S3 
•§ 


<§    S 


| 


w 
A 

< 
H 


§  g 
1.1 

GO      G" 
EH      o& 

« 

o 


^ 


^H  rH  i-H  i-H  (N 


T-H  rH  rH  ^H  CS|  (N 


T-H  T-H  i-l  ,-H  (M  C<J  CO 


s        I 

00 


i-t  i-H  ^H  rH  (N  CQ  CO 


7. 


S  u  ® 

1*1 


d 


I>O5T-HCOiOI>OlO'-il> 


,H  T-H  rH  rH  <M  (M  CO  CO  Tt<  »O  «5  I>  O5 


o  o»  S  t>»  o  w  8  1>»  o  c«i  «5  1»  o  «>  S  8  8 


226 


DIAGRAMS,  FORMULAS,  AND  TABLES. 


[Cn.  VI. 


TABLE  No.  20. 
MATERIALS  REQUIRED  FOR  ONE  CUBIC  YARD  OF  CONCRETE. 


Proportion  of  Mixture. 

Required  for  One  Cubic  Yard. 

Cement. 

Sand. 

Stone. 

Ratio: 
Mortar 

Cement, 
Barrels. 

Sand, 
Cubic  Yards. 

Stone, 
Cubic  Yards. 

Stone 

1 

2.0 

.70 

2.57 

0.39 

0.78 

1 

2.5 

.56 

2.29 

0.35 

0.87 

1 

3.0 

.47 

2.06 

0.31 

0.94 

1.5 

2.5 

.71 

2.05 

0.47 

0.78 

1.5 

3.0 

.60 

.85 

0.42 

0.84 

1.5 

3.5 

.51 

.72 

0.39 

0.91 

1.5 

4.0 

.44 

.57 

0.36 

0.96 

2.0 

3.0 

.72 

.70 

0.52 

0.77 

2.0 

3.5 

.62 

.57 

0.48 

0.83 

2.0 

4.0 

.54 

.46 

0.44 

0.89 

2.0 

4.5 

.48 

.36 

0.42 

0.93 

2.5 

4.0 

.64 

.35 

0.52 

0.82 

2.5 

4.5 

.57 

.27 

0.48 

0.87 

2.5 

5.0 

.51 

.19 

0.46 

0.91 

2.5 

5.5 

.46 

.13 

0.43 

0.94 

3 

4.5 

.66 

.18 

0.54 

0.81 

3 

5.0 

.60 

.11 

0.51 

0.85 

3 

5.5 

.54 

.06 

0.48 

0.89 

1 

3 

6.0 

.50 

.00 

0.46 

0.92 

1 

3 

6.5 

.46 

.96 

0.44 

0.95 

§  152.] 


TABLES. 


227 


l. 


TABLE  No.  21. 

STRENGTH  OF  FLOOR-SLABS. 
Calculated  for  M  =  \wl2;  for  M=-fawl*  multiply  given  span  lengths  by  1.12. 

/c-400  #=59 

/«=  12,000  p=.0056 


J 

£ 

•g^  g 

to  ° 

1 

Span  in  Feet  for  Given  Net  Loads  per  Square  Foot 

1  C 

2  * 

J 

£fe  d- 

tfJlJ 

11 

of  Floor  in  Pounds. 

gj 

_tf} 

°^Ji 
111 

111 

*o  <B  es 
-*j  & 

I8p8 

-2*0 

|§£ 

Bin's 

|9i 

•S^J 

50 

75 

100 

150 

200 

250 

300 

400 

500 

H 

H 

1 

§ 

£ 

2 

.0833 

1100 

24.2 

3.2 

2.7 

2.4 

2.1 

1.8 

1.6 

1.5 

2i 

.1167 

2200 

30.3 

4.2 

3.7 

3.3 

2.8 

2.5 

2.3 

2.1 

1.8 

3 

.150 

3600 

36.4 

5.3 

4.6 

4.2 

3.6 

3.2 

2.9 

2.6 

2.3 

3J 

.183 

5400 

42.5 

6.2 

5.5 

5.0 

4.3 

3.8 

3.5 

3.2 

2.8 

4 

1 

.200 

6400 

48.5 

6.6 

5.9 

5.4 

4.6 

4.1 

3.8 

3.5 

3.1 

2.8 

4A 

1 

.233 

8700 

54.6 

7.5 

6.7 

6.1 

5.3 

4.8 

4.3 

4.0 

3.5 

3.2 

5" 

1 

.267 

11400 

60.6 

8.3 

7.5 

6.8 

6.0 

5.3 

4.9 

4.5 

4.0 

3.6 

SJ 

1 

.300 

14400 

66.7 

9.1 

8.2 

7.6 

6.6 

6.0 

5.5 

5.1 

4.5 

4.1 

6 

U 

.317 

16000 

72.7 

9.3 

8.5 

7.8 

6.9 

6.2 

5.7 

5.3 

4.7 

4.3 

7 

If 

.383 

23500 

84.8 

10.8 

9.9 

9.2 

8.2 

7.4 

6.8 

6.4 

5.7 

5.1 

8 

U 

.450 

32400 

97.0 

12.1 

11.2 

10.5 

9.3 

8.5 

7,9 

7.4 

6.6 

6.0 

9 

If 

.500 

40000 

109.1 

12.9 

12.0 

11.3 

10.1 

9.3 

8.6 

8.0 

7.2 

6.6 

10 

11 

- 

.567 

51400 

121.3 

14.1 

13.2 

12.4 

11.2 

10.3 

9.6 

9.0 

8.1 

7.4 

12 

1* 

.700 

78400 

145.7 

16.4 

15.4 

14.6 

13.3 

12.3 

11.4 

10.8 

9.8 

9.0 

/c  =  400 
fs=  14,000 


#=54 
p=  .0043 


2 

.064 

1000 

24.2 

3.0 

2.6 

2.3 

2.0 

1.7 

1.6 

2* 

.090 

2000 

30.3 

4.1 

3.5 

3.2 

2.7 

2.4 

2.2 

2.0 

1.7 

3 

.116 

3300 

36.4 

5.0 

4.4 

4.0 

3.4 

3.0 

2.8 

2.5 

2.2 

2.0 

3J 

.141 

4900 

42.5 

5.9 

5.3 

4.8 

4.1 

3.7 

3.3 

3.1 

2.7 

2.4 

4 

1 

.154 

5800 

48.5 

6.3 

5.6 

5.1 

4.4 

4.0 

3.6 

3.3 

2.9 

2.6 

4* 

1 

.180 

7900 

54.6 

7.1 

6.4 

5.8 

5.1 

4.5 

•  4.2 

3.8 

3.4 

3.1 

5 

1 

.206 

10400 

60.6 

7.9 

7.1 

6.5 

5.7 

5.1 

4.7 

4.4 

3.9 

3.5 

5J 

1 

.231 

13100 

66.6 

8.6 

7.9 

7.2 

6.4 

5.7 

5.2 

4.9 

4.3 

3.9 

6 

M 

.244 

14600 

72.6 

8.9 

8.1 

7.5 

6.6 

6.0 

5.5 

5.1 

4.5 

4.1 

7 

ii 

.296 

21400 

84.7 

10.3 

9.5 

8.8 

7.8 

7.1 

6.5 

6.1 

5.4 

4.9 

8 

ij 

.347 

29500 

96.9 

11.6 

10.7 

10.0 

8.9 

8.1 

7.5 

7.0 

6.3 

5.7 

9 

i- 

i 

.386 

36400 

109.0 

12.4 

11.5 

10.8 

9.7 

8.9 

8.2 

7.7 

6.9 

6.3 

10 

i 

.437 

46800 

121.1 

13.5 

12.6 

11.9 

10.7 

9.8 

9.2 

8.6 

7.7 

7.1 

12 

i 

I 

.540 

71400 

145.4 

15.6 

14.7 

13.9 

12.7 

11.7 

11.0 

10.3 

9.3 

8.6 

228 


DIAGRAMS,  FORMULAS,  AND  TABLES. 


[Cn.  VI. 


TABLE  No.  21 — Continued. 
STRENGTH  OF  FLOOR-SLABS. 

Calculated  for  M=^wl2;  for  M  =  ^ivl2  multiply  given  span  lengths  by  1.12. 
/c  =  400  R=52 

js=  15,000  p=.0038 


3. 


ll 

i 

"o-g  fl 

i2 

& 

•Span  in  Feet  for  Given  Net  Loads  per  Square  Foot 

u  ^ 

o 

m  O     . 

-S    O    03 

o3"o 

of  Floor  in  Pounds. 

c  c 

ol 

"o 
~ 

<i  f~to 

K^:2 

CO  0 

'""  JD 

00  .S 

T3  ft^" 

°&s 

*O  0) 

H| 

l| 

1 

111 

+S    W 

£  g.g 

slas 

1~ 

«  £. 

!3° 

£ 

-*^  t»-« 

|  rtCC 

'ScoS 

50 

75 

100 

150 

200 

250 

300 

400 

500 

H 

H 

PH 

§ 

^ 

2 

.057 

1000 

24.1 

2.9 

2.6 

2.3 

1.9 

1.7 

1.5 

2| 

.080 

1900 

30.3 

4.0 

3.5 

3.1 

2.6 

2.3 

2.1 

1.9 

1.7 

1.5 

3 

.103 

3100 

36.4 

4.9 

4.3 

3.9 

3.3 

3.0 

2.7 

2.5 

2.2 

2.0 

3* 

< 

.126 

4700 

42.5 

5.8 

5.2 

4.7 

4.0 

3.6 

3.3 

3.0 

2.7 

2.4 

4 

1 

.137 

5600 

48.5 

6.1 

5.5 

5.0 

4.3 

3.9 

3.5 

3.3 

2.9 

2.6 

.j  i, 

1 

.160 

7600 

54.6 

6.9 

6.3 

5.7 

5.0 

4.5 

4.1 

3.8 

3.3 

3.0 

5 

1 

.183 

9900 

60.5 

7.7 

7.0 

6.4 

5.6 

5.0 

4.6 

4.3 

3.8 

3.4 

5i 

1 

.206 

12600 

66.5 

8.5 

7.7 

7.1 

6.2 

5.6 

5.1 

4.8 

4.2 

3.8 

6 

H 

.217 

14000 

72.5 

8.7 

8.0 

7.4 

6.5 

5.8 

5.4 

5.0 

4.4 

4.0 

7 

U 

.263 

20500 

83.6 

10.1 

9.3 

8.6 

7.7 

6.9 

6.4 

6.0 

5.3 

4.8 

8 

i. 

.309 

28300 

96.8 

11.4 

10.5 

9.8 

8.7 

8.0 

7.4 

6.9 

6.2 

5.6 

9 

l* 

.343 

34900 

108.9 

12.1 

11.3 

10.6 

9.5 

8.7 

8.1 

7.6 

6.8 

6.2 

10 

i. 

.389 

44800 

120.9 

13.2 

12.4 

11.6 

10.5 

9.7 

9.0 

8.4 

7.6 

6.9 

12 

r 

.480 

68400 

145.2 

15.3 

14.4 

13.7 

12.5 

11.5 

10.8 

10.1 

9.2 

8.4 

4. 


/c  =  400 
/8=16,000 


#=50 
p=  .0034 


2 

.051 

900 

24.1 

2.9 

2.5 

2.2 

1.9 

1.6 

1.5 

2.1 

\ 

.072 

1800 

30.3 

3.9 

3.4 

3.1 

2.6 

2.3 

2.1 

1.9 

1.7 

1.5 

3 

.092 

3000 

36.4 

4.8 

4.2 

3.8 

3.3 

2.9 

2.6 

2.4 

2.1 

1.9 

\ 

.112 

4500 

42.5 

5.7 

5.0 

4.6 

3.9 

3.5 

3.2 

2.9 

2.6 

2.3 

4 

1 

.123 

5400 

48.5 

6.0 

5.3 

4.9 

4.2 

3.8 

3.4 

3.2 

2.8 

2.5 

4  1 

1 

.143 

7300 

54.6 

6.8 

6.1 

5.6 

4.8 

4.3 

4.0 

3.7 

3.2 

2.9 

5 

1 

.164 

9500 

60.5 

7.5 

6.8 

6.2 

5.4 

4.9 

4.5 

4.2 

3.7 

3.3 

5i 

1 

.184 

12100 

66.5 

8.2 

7.5 

6.9 

6.0 

5.4 

5.0 

4.6 

4.1 

3.7 

6 

H 

.194 

13400 

72.5 

8.5 

7.7 

7.2 

6.3 

5.7 

5.2 

4.9 

4.3 

3.9 

7 

If 

.235 

19700 

83.5 

9.8 

9.0 

8.4 

7.4 

6.8 

6.2 

5.8 

5.2 

4.7 

8 

11 

.276 

27100 

96.7 

11.0 

10.2 

9.5 

8.5 

7.7 

7.2 

6.7 

6.0 

5.4 

9 

U 

f. 

.307 

33500 

108.7 

11.8 

11.0 

10.3 

9.2 

8.4 

7.8 

7.3 

6.6 

6.0 

10 

i 

r 

.348 

43000 

120.7 

12.9 

12.0 

11.3 

10.2 

9.4 

8.7 

8.2 

7.4 

6.7 

12 

ij 

.430 

65600 

145.0 

14.9 

14.0 

13.3 

12.1 

11.2 

10.4 

9.8 

8.9 

8.2 

§  152.] 


TABLES. 


229 


TABLE  No.  21 — Continued. 
STRENGTH  OF  FLOOR-SLABS. 


Calculated  for  M=%wl2',   for  M  =  ^wl2  multiply  given  span  lengths  by  1.12. 

/c=400                                    #=46 

5'                                 /«=  18,000                              p=.0028 

il 

§ 

*  j 

1*8 

I 

Span  in  Feet  for  Given  Net  Loads  per  Square  Foot 

00 

o     S 

tlcf 

ll| 

JS§ 

of  Floor  in  Pounds. 

o    _ 

C3 

^    tn  ^* 

<L—    — 

&H 

111" 

111 

§  §53 

||, 

Jhg 

•-  002 

£02  "o 

1S-S 

'Sosfh^ 

50 

75 

100 

150 

200 

250 

300 

400 

500 

H 

H 

S 

2 

| 

.042 

900 

24.1 

2.8 

2.4 

2.1 

1.8 

1.6 

1.4 

1.3 

21 

i 

.058 

1700 

30.3 

3.7 

3.3 

2.9 

2.5 

2.2 

2.0 

1.8 

1.6 

1.4 

3 

i 

.075 

2800 

36.4 

4.6 

4.1 

3.7 

3.2 

2.8 

2.5 

2.3 

2.1 

1.8 

3* 

i 

.092 

4200 

42.5 

5.5 

4.9 

4.4 

3.8 

3.4 

3.1 

2.8 

2.5 

2.3 

4 

i 

.100 

5000 

48.5 

5.8 

5.2 

4.7 

4.1 

3.6 

3.3 

3.1 

2.7 

2.4 

41 

i 

.117 

6700 

54.6 

6.6 

5.9 

5.4 

4.7 

4.2 

3.8 

3.6 

3.1 

2.8 

5 

i 

.133 

8800 

60.4 

7.3 

6.6 

6.1 

5.3 

4.7 

4.3 

4.0 

3.6 

3.2 

H 

1 

.150 

11100 

66.4 

8.0 

7.3 

6.7 

5.9 

5.3 

4.8 

4.5 

4.0 

3.6 

6 

ij 

.158 

12400 

72.4 

8.2 

7.5 

6.9 

6.1 

5.5 

5.1 

4.7 

4.2 

3.8 

7 

i^ 

.192 

18200 

83.4 

9.5 

8.8 

8.1 

7.2 

6.5 

6.0 

5.6 

5.0 

4.5 

8 

if 

.225 

25100 

96.6 

10.7 

9.9 

9.3 

8.3 

7.5 

6.9 

6.5 

5.8 

5.3 

9 

ii 

.250 

31000 

108.6 

11.4 

10.6 

10.0 

8.9 

8.2 

7.6 

7.1 

6.4 

5.8 

10 

ii 

.283 

39800 

120.6 

12.5 

11.6 

11.0 

9.9 

9.1 

8.5 

7.9 

7.1 

6.5 

12 

ij 

.350 

60700 

144.9 

14.4 

13.6 

12.9 

11.7 

10.8 

10.1 

9.5 

8.6 

7.9 

/c  =  500                                  #=84 

/*=  12,000                              p=.0080 

2 

<£ 

.120 

1600 

24.3 

3.8 

3.2 

2.9 

2.4 

2.2 

1.9 

1.8 

1.6 

1.4 

24 

1 

4  ' 

.168 

3100 

30.4 

5.1 

4.4 

4.0 

3.4 

3.0 

2.7 

2.5 

2.2 

-2.0 

3 

4 

.216 

5100 

36.5 

6.3 

5.5 

5.0 

4.3 

3.8 

3.4 

3.2 

2.8 

2.5 

31 

i 

.264 

7600 

42.7 

7.4 

6.6 

6.0 

5.1 

4.6 

4.2 

3.8 

3.4 

3.1 

4 

1 

.312 

9100 

48.7 

7.8 

7.0 

6.4 

5.5 

4.9 

4.5 

4.2 

3.7 

3.3 

41 

1 

.337 

12300 

54.8 

8.9 

8.0 

7.3 

6.3 

5.7 

5.2 

4.8 

4.3 

3.8 

5 

1 

.385 

16100 

60.9 

9.8 

8.9 

8.2 

7.1 

6.4 

5.9 

5.5 

4.8 

4.4 

5| 

1 

.433 

20400 

67.0 

10.8 

9.8 

9.0 

7.9 

7.1 

6.5 

6.1 

5.4 

4.9 

6 

If 

.457 

22700 

73.1 

11.1 

10.1 

9.4 

8.2 

7.4 

6.8 

6.4 

5.6 

5.1 

7 

if 

.553 

33300 

85.3 

12.8 

11.8 

10.9 

9.7 

8.8 

8.1 

7.6 

6.7 

6.1 

8 

If 

.649 

45800 

97.6 

14.4 

13.3 

12.4 

11.1 

10.1 

9.4 

8.8 

7.8 

7.1 

9 

14 

.721 

56600 

109.7 

15.4 

14.3 

13.4 

12.1 

11.0 

10.2 

9.6 

8.6 

7.9 

10 

if 

.817 

72700 

122.0 

16.8 

15.7 

14.8 

13.3 

12.3 

11.4 

10.7 

9.6 

8.8 

12 

11 

1.010 

110900 

146.5 

19.4 

18.3 

17.3 

15.8 

14.6 

13.6 

12.9 

11.6 

10.7 

230 


DIAGRAMS,  FORMULAS,  AND  TABLES. 


[Cn.  VI. 


TABLE  No.  21 — Continued. 

STRENGTH  OF  FLOOR-SLABS. 
Calculated  for  M=  \wl2;   for  M  =  -Il^wP  multiply  given  span  lengths  by  1.12. 

A;  =  500  R=ll 

fa=  14,000  p=.0062 


7. 


i! 

il 

iFj 

fi. 

a> 
ft 
,0-^ 

Span  in  Feet  for  Given  Net  Loads  per  Square  Foot 
of  Floor  in  Pounds. 

J^  >~H 

*oj2  w 

1^02 

"  •—  < 

02  ^ 

<g^M 

|iU 

*O  a)  tf 

*o  v 

~02 

||| 

l*s 

slj 

1° 

gSoQ 

Is"8 

|sl 

|<ES 

50 

75 

100 

150 

200 

250 

300 

400 

500 

2 

| 

.094 

1400 

24.2 

3.6 

3.1 

2.8 

2.4 

2.1 

1.9 

1.7 

1.5 

1.4 

2i 

| 

.131 

2800 

30.3 

4.8 

4.2 

3.8 

3.2 

2.9 

2.6 

2.4 

2.1 

1.9 

3 

.168 

4700 

36.4 

6.0 

5.3 

4.8 

4.1 

3.6 

3.3 

3.0 

2.7 

2.4 

3| 

I 

.206 

7000 

42.5 

7.1 

6.3 

5.7 

4.9 

4.4 

4.0 

3.7 

3.3 

2.9 

4 

1 

.224 

8300 

48.5 

7.5 

6.7 

6.1 

5.3 

4.7 

4.3 

4.0 

3.5 

3.2 

4| 

1 

.262 

11300 

54.6 

8.5 

7.6 

7.0 

6.1 

5.5 

5.0 

4.6 

4.1 

3.7 

5 

1 

.299 

14800 

60.7 

9.5 

8.5 

7.9 

6.8 

6.2 

5.7 

5.2 

4.6 

4.2 

5* 

1 

.336 

18700 

66.8 

10.4 

9.4 

8.7 

7.6 

6.9 

6.3 

5.9 

5.2 

4.7 

6 

11 

.355 

20900 

72.9 

10.5 

9.5 

8.8 

7.8 

7.0 

6.5 

6.0 

5.4 

4.8 

7 

if 

.430 

30600 

85.1 

12.2 

11.1 

10.3 

9.3 

8.3 

7.7 

7.2 

6.5 

5.8 

8 

l| 

.505 

42100 

97.4 

13.7 

12.6 

11.8 

10.6 

9.6 

9.0 

8.3 

7.5 

6.8 

9 

1| 

.561 

52000 

109.5 

14.8 

13.7 

12.9 

11.5 

10.6 

9.8 

9.2 

8.3 

7.6 

10 

l| 

.636 

66800 

121.7 

16.1 

15.0 

14.2 

12.8 

11.8 

11.0 

10.3 

9.3 

8.5 

12 

U 

.785 

102000 

145.9 

18.9 

17.5 

16.6 

15.1 

14.1 

13.1 

12.4 

11.2 

10.3 

8. 


/8=  15,000 


#-74 
p=.0056 


2 

.083 

1400 

24.2 

3.5 

3.1 

2.7 

2.3 

2.0 

1.8 

1.7 

1.5 

1.3 

2| 

.117 

2700 

30.3 

4.7 

4.1 

3.7 

3.2 

2.8 

2.5 

2.3 

2.0 

1.8 

3 

.150 

4500 

36.4 

5.9 

5.2 

4.7 

4.0 

3.6 

3.2 

3.0 

2.6 

2.4 

3£ 

.183 

6700 

42.5 

7.0 

6.2 

5.6 

4.8 

4.3 

3.9 

3.6 

3.2 

2.9 

4 

1 

.200 

8000 

48.5 

7.4 

6.6 

6.0 

5.2 

4.6 

4.2 

3.9 

3.4 

3.1 

4i 

1 

.233 

10900 

54.6 

8.3 

7.5 

6.9 

6.0 

5.3 

4.9 

4.5 

4.0 

3.6 

5 

1 

.267 

14200 

60.6 

9.2 

8.3 

7.7 

6.7 

6.0 

5.5 

5.1 

4.5 

4.1 

5| 

1 

.300 

18000 

66.7 

10.1 

9.2 

'8.5 

7.4 

6.7 

6.2 

5.7 

5.1 

4.6 

6 

H 

.317 

20100 

72.8 

10.4 

9.5 

8.8 

7.7 

7.0 

6.4 

6.0 

5.3 

4.8 

7 

H 

.383 

29400 

84.9 

12.0 

11.1 

10.3 

9.1 

8.3 

7.6 

7.1 

6.4 

5.8 

8 

li 

.450 

40500 

97.2 

13.5 

12.5 

11.7 

10.5 

9.5 

8.8 

8.2 

7.4 

6.7 

9 

n 

.500 

50000 

109.3 

14.5 

13.5 

12.6 

11.3 

10.4 

9.6 

9.0 

8.1 

7.4 

10 

u 

.567 

64200 

121.5 

15.8 

14.7 

13.9 

12.6 

11.5 

10.7 

10.1 

9.1 

8.3 

12 

11 

.700 

98000 

145.7 

18.2 

17.2 

16.3 

14.8 

13.7 

12.8 

12.1 

10.9 

10.0 

152.] 


TABLES. 


231 


TABLE  No.  21 — Continued. 
STRENGTH  OF  FLOOR-SLABS. 

Calculated  for  M=$wP',   for  M=^wP  multiply  given  span  lengths  by  1.12. 
/c  =  500  #=71 

fa=  16,000  p=.0050 


9. 


g 

il 

§ 

"o-g^ 

•I  ° 

a 

Span  in  Feet  for  Given  Net  Loads  per  Square  Foot 

03  "y 

iu  £Ja 

^fa    • 

($  °  " 

|| 

of  Floor  in  Pounds. 

*0^ 

«-i"S 

«l<? 

«-^'7 

'£•& 

l*"| 

1-1 

IOT 

^£ 

Q)   C  *"^ 

p 

o  £-M 

^3  oCQ 

|h 

1 

_M  QV2 

50 

75 

100 

150 

200 

250 

300 

400 

500 

2 

t 

.075 

1300 

24.2 

3.5 

3.0 

2.7 

2.3 

2.0 

1.8 

1.7 

1.4 

1.3 

21 

I 

.105 

2600 

30.3 

4.7 

4.1 

3.7 

3.1 

2.8 

2.5 

2.3 

2.0 

1.8 

3 

I 

.135 

4300 

36.4 

5.8 

5.1 

4.6 

4.0 

3.5 

3.2 

2.9 

2.6 

2.3 

31 

I 

.165 

6500 

42.5 

6.8 

6.1 

5.5 

4.8 

4.2 

3.8 

3.6 

3.1 

2.8 

4 

1 

.180 

7700 

48.5 

7.2 

6.4 

5.9 

5.1 

4.5 

4.1 

3.8 

3.4 

3.1 

4| 

1 

.209 

10500 

54.7 

8.2 

7.3 

6.7 

5.8 

5.2 

4.8 

4.4 

3.9 

3.5 

5 

1 

.239 

13700 

60.7 

9.1 

8.2 

7.5 

6.6 

5.9 

5.4 

5.0 

4.4 

4.0 

5| 

1 

.269 

17300 

66.6 

10.0 

9.1 

8.3 

7.3 

6.6 

6.0 

5.6 

5.0 

4.5 

6 

11 

.284 

19300 

72.7 

10.3 

9.4 

8.7 

7.6 

6.9 

6.3 

5.9 

5.2 

4.7 

7 

li 

.344 

28300 

84.8 

11.9 

10.9 

10.1 

9.0 

8.2 

7.5 

7.0 

6.3 

5.7 

8 

1| 

.404 

39000 

97.0 

13.4 

12.3 

11.5 

10.3 

9.4 

8.7 

8.1 

7.3 

6.6 

9 

11 

.449 

48100 

109.1 

14.3 

13.3 

12.4 

11.2 

10.2 

9.5 

8.9 

8.0 

7.3 

10 

If 

.509 

61800 

121.3 

15.6 

14.6 

13.7 

12.4 

11.4 

10.6 

9.9 

8.9 

8.2 

12 

if 

.629 

94400 

145.5 

17.9 

16.9 

16.0 

14.6 

13.5 

12.6 

11.9 

10.7 

9.9 

/c  =  500 
/«  =  18,000 


#=66 
p=.0041 


2 

. 

.061 

1200 

24.2 

3.3 

2.9 

2.6 

2.2 

1.9 

1.7 

1.6 

1.4 

1.3 

| 

.086 

2400 

30.3 

4.5 

3.9 

3.5 

3.0 

2.6 

2.4 

2.2 

1.9 

1.7 

3 

| 

.110 

4000 

36.4 

5.6 

4.9 

4.4 

3.8 

3.4 

3.0 

2.8 

2.5 

2.2 

1 

.135 

6000 

42.4 

6.6 

5.9 

5.3 

4.6 

4.0 

3.7 

3.4 

3.0 

2.7 

4 

1 

.147 

7200 

48.4 

7.0 

6.2 

5.7 

4.9 

4.4 

4.0 

3.7 

3.3 

2.9 

4.1 

1 

.172 

9700 

54.5 

7.8 

7.1 

6.5 

5.6 

5.1 

4.6 

4.3 

3.8 

3.4 

5 

1 

.196 

12700 

60.5 

8.8 

7.9 

7.3 

6.4 

5.7 

5.2 

4.8 

4.3 

3.9 

51 

1 

.221 

16100 

68.5 

9.5 

8.7 

8.0 

7.0 

6.1 

5.8 

5.4 

4.8 

4.3 

6 

Ij 

.233 

18000 

72.6 

9.8 

9.0 

8.3 

7.3 

6.6 

6.1 

5.7 

5.0 

4.6 

7 

.282 

26300 

84.7 

11.4 

10.4 

9.7 

8.6 

7.8 

7.2 

6.7 

6.0 

5.5 

8 

l| 

.331 

36300 

96.8 

12.8 

11.8 

11.0 

9.9 

9.0 

8.3 

7.8 

7.0 

6.4 

9 

li 

.368 

44800 

108.9 

13.6 

12,7 

11.9 

10.7 

9.8 

9.1 

8.5 

7.7 

7.0 

10 

if 

.417 

57500 

121.1 

14.9 

13.9 

13.1 

11.9 

10.9 

10.1 

9.5 

8.6 

7.9 

12 

H 

.515 

87700 

145.3 

17.1 

16.2 

15.4 

14.0 

13.0 

12.1 

11.4 

10.4 

9.5 

232 


DIAGRAMS,  FORMULAS,  AND  TABLES. 


[On.  VI. 


Calculated  for  M 
11. 


TABLE  No.  21— -Continued. 
STRENGTH  OF  FLOOR-SLABS. 

%wl2;   for  M  =  ^wl2  multiply  given  span  lengths  by  1.12. 
/c  =  600  #=110 

fs=  12,000  p=.0107 


1 

j 

•s  d 

|<S 

1 

Span  in  Feet  for  Given  Net  Loads  per  Square  Foot 

jj 

GO  pJ2 

1 

c3  o1""1 

84 

111 

11 
"o  g 

of  Floor  in  Pounds. 

!-! 

3  It 

'3  IS 

f  8-f 

1*J 

•I*® 

S  £•5 

I  is 

•5?o?i3 

50 

75 

100 

150 

200 

250 

300 

400 

500 

H 

H  ° 

tf 

jjj 

5 

2 

.161 

2100 

24.4 

4.3 

3.7 

3.3 

2.8 

2.5 

2.2 

2.0 

1.8 

1.6 

21 

.225 

4000 

30.5 

5.8 

5.0 

4.5 

3.8 

3.4 

3.1 

2.8 

2.5 

2.2 

3 

.289 

6700 

36.6 

7.2 

6.3 

5.7 

4.9 

4.3 

3.9 

3.6 

3.2 

2.9 

31 

.354 

10000 

42.9 

8.5 

7.5 

6.8 

5.8 

5.2 

4.7 

4.4 

3.8 

3.5 

4 

1 

.386 

11900 

48.9 

8.9 

8.0 

7.3 

6.3 

5.6 

5.1 

4.7 

4.2 

3.8 

1 

.450 

16200 

55.0 

10.1 

9.1 

8.3 

7.2 

6.5 

5.9 

5.5 

4.8 

4.4 

5 

1 

.514 

21200 

61.2 

11.2 

10.1 

9.3 

8.2 

7.3 

6.7 

6.2 

5.5 

5.0 

5£ 

1 

.579 

26800 

67.4 

12.3 

11.1 

10.3 

9.0 

8.1 

7.5 

6.9 

6.1 

5.6 

6 

1; 

• 

.611 

29800 

73.5 

12.7 

11.5 

10.7 

9.4 

8.5 

7.8 

7.3 

6.4 

5.8 

7 

' 

.739 

43700 

85.8 

14.6 

13.4 

12.5 

11.1 

10.1 

9.3 

8.7 

7.7 

7.0 

8 

H 

.868 

60300 

98.2 

16.4 

15.2 

14.2 

12.7 

11.6 

10.7 

10.0 

8.9 

8.2 

9 

li 

;. 

.964 

74400 

110.3 

17.5 

16.3 

15.3 

13.8 

12.6 

11.7 

10.9 

9.8 

9.0 

10 

ij 

1.093 

95500 

122.6 

19.2 

17.9 

16.9 

15.2 

14.0 

13.0 

12.2 

11.0 

10.1 

12 

M 

1.350 

145800 

147.2 

22.2 

20.9 

19.8 

18.0 

16.7 

15.6 

14.7 

13.3 

12.2 

±2. 


/c=600 
/«=  14,000 


p=.0084 


2 

, 

.126 

1900 

24.3 

4.1 

3.6 

3.2 

2.7 

2.4 

2.2 

2.0 

1.7 

1.6 

21 

1 
•4 

.176 

3800 

30.4 

5.6 

4.9 

4.4 

3.7 

3.3 

3.0 

2.7 

2.4 

2.2 

3 

1 

.226 

6200 

36.5 

6.9 

6.1 

5.5 

4.7 

4.2 

3.8 

3.5 

3.1 

2.8 

31 

i 
¥ 

.277 

9300 

42.7 

8.2 

7.2 

6.6 

5.7 

5.0 

4.6 

4.2 

3.7 

3.4 

4 

1 

.302 

11000 

48.7 

8.6 

7.7 

7.0 

6.1 

5.4 

5.0 

4.6 

4.0 

3.7 

41 

1 

.352 

15000 

54.8 

9.8 

8.8 

8.0 

7.0 

6.3 

5.7 

5.3 

4.7 

4.3 

5 

1 

.403 

19600 

60.9 

10.8 

9.8 

9.0 

7.9 

7.1 

6.5 

6.0 

5.3 

4.8 

51 

1 

.453 

24800 

67.0 

11.9 

10.8 

10.0 

8.7 

7.9 

7.2 

6.7 

6.0 

5.4 

6 

II 

.478 

27700 

73.1 

12.2 

11.2 

10.3 

9.1 

8.2 

7.5 

7.0 

6.2 

5.7 

7 

H 

.579 

40600 

85.4 

14.1 

13.0 

12.1 

10.7 

9.7 

9.0 

8.4 

7.5 

6.8 

8 

H 

.680 

55900 

97.8 

15.9 

14.7 

13.7 

12.2 

11.2 

10.3 

9.7 

8.6 

7.9 

9 

11 

.755 

69000 

109.8 

16.9 

15.7 

14.8 

13.3 

12.2 

11.3 

10.6 

9.5 

8.7 

10 

if 

.856 

88600 

122.0 

18.5 

17.3 

16.3 

14.7 

13.5 

12.6 

11.8 

10.6 

9.7 

12 

ij 

1.057 

135300 

146.4 

21.4 

20.1 

19.1 

17.4 

16.1 

15.0 

14.2 

12.8 

11.8 

§  152.] 


TABLES. 


233 


Calculated  for  M 
13. 


TABLE  No.  21 — Continued. 

STRENGTH  OF  FLOOR-SLABS. 

wl2',  for  M  =  ^wl*  multiply  given  span  lengths  by  1.12. 

/c=600  #=98 

/.  =  15,000  p=.0075 


08  J3 

il 

"O-ta   C 

•*>  *o 

1g« 

ft 

Span  in  Feet  for  Given  Net  Loads  per  Square  Foot 
of  Floor  in  Pounds 

o    , 

"°'S  c 

«!   ^M__ 

**"  '-'a 

^£ 

g| 

Sii 

Sll 

l|2 

•g  o3    . 

1° 

ill 

gitt'o 

III 

'Sc«5 

50 

75 

100 

150 

200 

250 

300 

400 

500 

EH 

H 

« 

s 

^ 

2 

a 

.112 

1800 

24.2 

4.1 

3.5 

3.2 

2.7 

2.3 

2.1 

1.9 

1.7 

1.5 

2* 

| 

.157 

3600 

30.4 

5.5 

4.8 

4.3 

3.7 

3.2 

2.9 

2.7 

2.4 

2.1 

3 

.202 

6000 

36.5 

6.8 

6.0 

5.4 

4.6 

•  4.1 

3.7 

3.4 

3.0 

2.7 

3* 

! 

.247 

8900 

42.7 

8.0 

7.1 

6.5 

5.6 

5.0 

4.5 

4.2 

3.7 

3.3 

4 

.270 

10600 

48.7 

8.5 

7.6 

6.9 

6.0 

5.3 

4.9 

4.5 

4.0 

3.6 

4* 

.315 

14500 

54.8 

9.6 

8.6 

7.9 

6.9 

6.2 

5.6 

5.2 

4.6 

4.2 

5 

.360 

18900 

60.9 

10.7 

9.6 

8.8 

7.7 

7.0 

6.4 

5.9 

5.2 

4.7 

5* 

.405 

23900 

67.0 

11.7 

10.6 

9.8 

8.6 

7.7 

7.1 

6.6 

5.8 

5.3 

6 

i 

.427 

26700 

73.1 

12.0 

11.0 

10.1 

8.9 

8.1 

7.4 

6.9 

e'.i 

5.6 

7 

^ 

.517 

39100 

85.5 

13.9 

12.8 

11.9 

10.5 

9.6 

8.8 

8.2 

7.3 

6.7 

8 

i 

.607 

53800 

97.9 

15.6 

14.4 

13.5 

12.1 

11.0 

10.2 

9.5 

8.5 

7.8 

9 

it 

.675 

66400 

109.6 

16.7 

15.5 

14.6 

13.1 

12.0 

11.1 

10.4 

9.4 

8.5 

10 

i  j 

.765 

85300 

121.8 

18.2 

17.0 

16.0 

14.5 

13.3 

12.4 

11.6 

10.4 

9.6 

12 

it 

.945 

130200 

146.2 

21.1 

19.8 

18.8 

17.1 

15.8 

14.8 

14.0 

12.6 

11.6 

/c=600 
/«=  16,000 


#=95 


2 

.101 

1800 

24.2 

4.0 

3.5 

3.1 

2.6 

2.3 

2.1 

1.9 

1.7 

1.5 

2* 

.142 

3500 

30.3 

5.4 

4.7 

4.2 

3.6 

3.2 

2.9 

2.7 

2.3 

2.1 

3 

.182 

5800 

36.4 

6.7 

5.9 

5.3 

4.5 

4.0 

3.7 

3.4 

3.0 

2.7 

3| 

.223 

8600 

42.6 

7.9 

7.0 

6.3 

5.5 

4.9 

4.4 

4.1 

3.6 

3.3 

4 

1 

.243 

10300 

48.6 

8.3 

7.4 

6.8 

5.9 

5.2 

4.8 

4.4 

3.9 

3.5 

41 

1 

.284 

14000 

54.7 

9.4 

8.5 

7.8 

6.7 

6.0 

5.5 

5.1 

4.5 

4.1 

5 

1 

.324 

18300 

60.8 

10.5 

9.5 

8.7 

7.6 

6.8 

6.3 

5.8 

5.1 

4.7 

5J 

1 

.365 

23100 

66.9 

11.5 

10.4 

9.6 

8.4 

7.6 

7.0 

6.5 

5.7 

5.2 

6 

li 

.385 

25700 

73.0 

11.8 

10.7 

9.9 

8.7 

7.9 

7.3 

6.8 

6.0 

5.5 

7 

ii 

.466 

37700 

85.2 

13.6 

12.5 

11.6 

10.3 

9.4 

8.7 

8.1 

7.2 

6.5 

8 

i.i 

.547 

52000 

97.7 

15.3 

14.2 

13.2 

11.8 

10.8 

10.0 

9.4 

8.3 

7.6 

9 

il 

- 

.608 

64200 

109.4 

16.4 

15.2 

14.3 

12.8 

11.8 

10.9 

10.2 

9.2 

8.4 

10 

i\ 

- 

.689 

82400 

121.6 

17.9 

16.7 

15.7 

14.2 

13.1 

12.2 

11.4 

10.2 

9.4 

12 

\\ 

- 

.851 

125800 

146.2 

20.6 

19.4 

18.4 

16.8 

15.5 

14.5 

13.7 

12.4 

11.4 

234 


DIAGRAMS,  FORMULAS,  AND  TABLES. 


[On  VI. 


TABLE  No.  21 — Continued. 
STRENGTH  OF  FLOOR-SLABS. 

Calculated  for  M=$wl2;   for  M  =  ^wP  'multiply  given  span  lengths  by  1.12. 
/c  =  600  #-89 

/s-18:000  p 


15. 


ll 

1 

1*s 

4*8 

a 

Span  in  Feet  for  Given  Net  Loads  per  Square  Foot 

V  0 

O  j.     O 

go   . 

^  O  03 

cd  O 

of  Floor  in  Pounds. 

r§HH 

oil 

5     tu 

^fc;£ 

IJ 

jj-a 

i-s- 

TJ  §Lfl 

3  s,^ 

^ 
|-s 

Jfl 

Jlf 

1  11 

I  Is 

50 

75 

100 

150 

200 

2.50 

300 

400 

500 

2 

i 

.083 

1700 

24.1 

3.9 

3.3 

3.0 

2.5 

2.2 

2.0 

1.8 

1.6 

1.4 

2J 

i 

.117 

3300 

30.3 

5.2 

4.5 

4.1 

3.5 

3.1 

2.8 

2.6 

2.2 

2.0 

3 

t 

.150 

5400 

36.4 

6.5 

5.7 

5.1 

4.4 

3.9 

3.5 

3.3 

2.9 

2.6 

3J 

i 

.183 

8100 

42.6 

7.6 

6.8 

6.1 

5.3 

4.7 

4.3 

4.0 

3.5 

3.1 

4 

i 

.200 

9600 

48.6 

8.1 

7.2 

6.6 

5.7 

5.1 

4.6 

4.3 

3.8 

3.4 

4£ 

i 

.233 

13100 

54.7 

9.1 

8.2 

7.5 

6.5 

5.8 

5.3 

4.9 

4.4 

4.0 

5 

i 

.267 

17100 

60.7 

10.1 

9.2 

8.4 

7.4 

6.6 

6.1 

5.6 

5.0 

4.5 

5* 

i 

.300 

21600 

66.8 

11.1 

10.1 

9.3 

8.2 

7.4 

6.7 

6.3 

5.6 

5.0 

6 

H 

.317 

24100 

72.8 

11.4 

10.4 

9.6 

8.5 

7.7 

7.1 

6.6 

5.8 

5.3 

7 

4 

.383 

35300 

85.2 

13.2 

12.1 

11.3 

10.0 

9.1 

8.4 

7.8 

7.0 

6.3 

8 

H 

.450 

48600 

97.5 

14.8 

13.7 

12.8 

11.4 

10.4 

9.6 

9.0 

8.1 

7.4 

9 

li 

.500 

60000 

109.2 

15.9 

14.7 

13.8 

12.4 

11.4 

10.6 

9.9 

8.9 

8.1 

10 

1$ 

.567 

77100 

121.3 

17.3 

16.2 

15.2 

13.8 

12.6 

11.8 

11.0 

9.9 

9.1 

12 

ii 

.700 

117600 

145.7 

20.0 

18.8 

17.8 

16.3 

15.1 

14.1 

13.3 

12.0 

11.0 

/c=700 
/.- 12,000 


72=138 


2 

.204 

2600 

24.5 

4.8 

4.2 

3.2 

3.1 

2.8 

2.5 

2.3 

2.0 

1.& 

2i 

.286 

5100 

30.7 

6.5 

5.8 

5.1 

4.3 

3.8 

3.5 

3.2 

2.8 

2.5 

3 

.368 

8400 

36.9 

8.0 

7.0 

6.4 

5.5 

4.8 

4.4 

4.1 

3.6 

3.2 

3| 

.449 

12500 

43.2 

9.4 

8.4 

7.6 

6.6 

5.8 

5.3 

4.9 

4.3 

3.9 

4 

.490 

14900 

49.2 

10.0 

8.9 

8.1 

7.0 

6.3 

5.7 

5.3 

4.7 

4.2 

4* 

.572 

20300 

55.3 

11.3 

10.2 

9.3 

8.1 

7.3 

6.6 

6.2 

5.4 

4.9 

5 

.654 

26500 

61.5 

12.5 

11.3 

10.4 

9.1 

8.2 

7.5 

7.0 

6.2 

5.6 

5* 

.735 

33500 

67.8 

13.8 

12.5 

11.5 

10.1 

9.1 

8.4 

7.8 

6.9 

6.3 

6 

I 

.776 

37400 

73.9 

14.1 

12.9 

11.9 

10.5 

9.5 

8.8 

8.2 

7.2 

6.6 

7 

.939 

54700 

86.2 

16.3 

15.0 

14.0 

12.4 

11.3 

10.4 

9.7 

8.7 

7  Q- 

8 

ll 

1.103 

75400 

98.7 

18.3 

17.0 

15.9 

14.2 

12.9 

12.0 

11.2 

10.0 

9^1 

9 

IJ 

1.225 

93100 

110.  Q 

19.6 

18.2 

17.1 

15.4 

14.1 

13.1 

12.3 

11.0 

10.1 

10 

If 

1.389 

119600 

123.3 

21.4 

20.0 

18.9 

17.1 

15.7 

14.6 

13.7 

12.3 

11.3 

12 

if 

1.716 

182500 

148.1 

24.7 

23.2 

22.1 

20.1 

18.7 

17.5 

16.3 

14.8 

13.7 

§  152.] 


TABLES. 


235 


TABLE  No.  21. — Continued. 

STRENGTH  OF  FLOOR-SLABS. 

Calculated  for  M=|wZ2;  for  M  =  ^wl2  multiply  given  span  lengths  by  1.12; 
/c=700  72=11:9 

fa=  14,000  p=.0107 


17. 


sl 

i 

«*-                    Q 

IsJ 

£, 

Span  in  Feet  for  Given  Net  Loads  per  Square  Foot 

•^3 

oj  o 

^J 

o 

r;  X 

ii* 

o3  "o 

of  Floor  in  Pounds. 

0  C 

**H     ^ 

_C 

^  r-^  O* 

l~ 

O-3 

~ 

•^  fa     . 

«*-  1.,1-1 

M  0 

1« 

"3 

||| 

ill 

-S 

-*j"o 

•»  £ 

i 

|g*S 

|S-s 

•-(15 

50 

75 

100 

150 

200 

250 

300 

400 

500 

1 

P 

I 

£ 

2 

4 

.161 

2400 

24.4 

4.7 

4.0 

3.6 

3.0 

2.7 

2.4 

2.2 

1.9 

1.7 

2i 

.225 

4700 

30.6 

6.3 

5.5 

4.9 

4.2 

3.7 

3.4 

3.1 

2.7 

2.4 

3 

.289 

7800 

36.7 

7.8 

6.8 

6.2 

5.3 

4.7 

4.3 

3.9 

3.5 

3.1 

3* 

.354 

11700 

43.0 

9.2 

8.1 

7.4 

6.4 

5.7 

5.2 

4.8 

4.2 

3.8 

4 

1 

.386 

13900 

48.9 

9.7 

8.6 

7.9 

6.8 

6.1 

5.6 

5.1 

4.5 

4.1 

1 

.450 

18900 

55.0 

11.0 

9.8 

9.0 

7.8 

7.0 

6.4 

5.9 

5.3 

4.8 

5 

1 

.514 

24700 

61.1 

12.2 

11.0 

10.1 

8.8 

7.9 

7.3 

6.7 

6.0 

5.4 

5* 

1 

.579 

31200 

67.4 

13.3 

12.1 

11.2 

9.8 

8.8 

8.1 

7.5 

6.7 

6.0 

6 

11 

.611 

34800 

73.5 

13.7 

12.5 

11.5 

10.2 

9.2 

8.5 

7.9 

7.0 

6.4 

7 

H 

1 

.739 

51000 

85.7 

15.8 

14.6 

13.5 

12.0 

10.9 

10.1 

9.4 

8.4 

7.0 

8 

U 

.868 

70300 

98.1 

17.8 

16.5 

15.4 

13.7 

12.5 

11.6 

10.9 

9.7 

8.9 

9 

if 

.964 

86800 

110.3 

19.0 

17.7 

16.6 

14.9 

13.7 

12.7 

11.9 

10.6 

9.7 

10 

i^ 

1.093 

111500 

122.6 

20.8 

19.4 

18.3 

16.5 

15.2 

14.1 

13.3 

11.9 

10.9 

12 

if 

1.350 

170100 

147.3 

24.0 

22.6 

21.5 

19.6 

18.1 

16.9 

15.9 

14.4 

13.2 

#-124 


15,000 


2 

.144 

2300 

24.4 

4.6 

4.0 

3.5 

3.0 

2.6 

2.4 

2.2 

1.9 

1.7 

2J 

.202 

4600 

30.6 

6.1 

5.4 

4.8 

4.1 

3.6 

3.3 

3.0 

2.7 

2.4 

3 

.259 

7600 

36.7 

7.6 

6.7 

6.1 

5.2 

4.6 

4.2 

3.9 

3.4 

3.1 

3i 

l 

.317 

11300 

42.9 

9.0 

8.0 

7.3 

6.3 

5.6 

5.1 

4.7 

4.1 

3.7 

4 

1 

.346 

13400 

48.8 

9.5 

8.5 

7.8 

6.7 

6.0 

5.6 

5.1 

4.5 

4.0 

4i 

1 

.404 

18300 

55.0 

10.8 

9.7 

8.9 

7.7 

6.9 

6.3 

5.9 

5.2 

4.7 

5" 

1 

.461 

23900 

61.1 

12.0 

10.8 

9.9 

8.7 

7.8 

7.1 

6.6 

5.9 

5.3 

5£ 

1 

.519 

30200 

67.3 

13.1 

11.9 

11.0 

9.6 

8.7 

8.0 

7.4 

6.6 

6.0 

6 

H 

.548 

33700 

73.3 

13.5 

12.3 

11.4 

10.0 

9.1 

8.3 

7.7 

6.9 

6.3 

7 

If 

.663 

49300 

85.5 

15.6 

14.3 

13.4 

11.8 

10.7 

9.9 

9.2 

8.2 

7.5 

8 

U 

.778 

68000 

97.9 

17.5 

16.2 

15.1 

13.5 

12.3 

11.4 

10.7 

9.5 

8.7 

9 

if 

.865 

83900 

110.1 

18.7 

17.4 

16.3 

14.7 

13.4 

12.5 

11.7 

10.5 

9.6 

10 

if 

.980 

107800 

122.3 

20.4 

19.1 

18.0 

16.2 

14.9 

13.9 

13.0 

11.7 

10.7 

12 

i| 

i 

1.210 

164500 

146.9 

23.6 

22.2 

21.1 

19.2 

17.8 

16.6 

15.7 

14.2 

13.0 

236 


DIAGRAMS,  FORMULAS,  AND  TABLES. 


[Cn.  VI. 


TABLE  No.  21 — Continued. 

STRENGTH  OF  FLOOR-SLABS. 

Calculated  for  M  =  $wl2;   for  M  =  ^wP  multiply  given  span  lengths  by  1.12. 
/c=700  fl=120 

/8=  16,000  p 


19. 


[  Thickness 
Slab,  Inches. 

"slf 

iS 

1 

«*-  j-,1""1 

|8| 

a 
5l 

Span  in  Feet  for  Given  Net  Loads  per  Square  Foot 
of  Floor  in  Pounds. 

s<s 

ill 

202*0 

jfs 

•Ioift3 

50 

75 

100 

150 

200 

250 

300 

400 

500 

H 

H 

K 

£ 

2 

.130 

2300 

24.3 

4.5 

3.9 

3.5 

2.9 

2.6 

2.3 

2.1 

1.9 

1.7 

21 

.182 

4400 

30.5 

6.1 

5.3 

4.8 

4.0 

3.6 

3.2 

3.0 

2.6 

2.4 

3 

.234 

7300 

36.6 

7.5 

6.6 

6.0 

5.1 

4.5 

4.1 

3.8 

3.3 

3.0 

31 

.286 

10900 

42.9 

8.9 

7.9 

7.2 

6.2 

5.5 

5.0 

4.6 

4.0 

3.7 

4 

.312 

13000 

48.8 

8.4 

8.4 

7.6 

6.6 

5.9 

5.4 

5.0 

4.4 

4.0 

41 

.364 

17700 

54.9 

10.6 

9.5 

8.7 

7.6 

6.8 

6.2 

5.8 

5.1 

4.6 

5 

.416 

23100 

61.0 

11.8 

10.6 

9.8 

8.5 

7.7 

7.0 

6.5 

5.8 

5.2 

51 

.468 

29200 

67.2 

12.9 

11.7 

10.8 

9.5 

8.5 

7.8 

7.3 

6.5 

5.9 

6 

i 

.494 

32600 

73.2 

13.3 

12.1 

11.2 

9.9 

8.9 

8.2 

7.6 

6.8 

6.2 

| 

.598 

47700 

85.4 

15.4 

14.1 

13.1 

11.6 

10.6 

9.8 

9.1 

8.1 

7.4 

8 

"    I 

• 

.702 

65800 

97.7 

17.3 

15.9 

14.9 

13.3 

12.1 

11.2 

10.5 

9.4 

8.6 

9 

I 

1 

.780 

81200 

109.9 

18.4 

17.1 

16.1 

14.4 

13.2 

12.3 

11.5 

10.3 

9.4 

10 

1 

.884 

104300 

122.1 

20.1 

18.8 

17.7 

16.0 

14.7 

13.7 

12.9 

11.6 

10.6 

12 

1.092 

159200 

146.6 

23.2 

21.9 

20.8 

18.9 

17.5 

16.4 

15.4 

14.0 

12.8 

20. 


/c=700 
fa=  18,000 


#=113 
p=.0072 


2 

.107 

2100 

24.3 

4.4 

3.8 

3.4 

2.8 

2.5 

2.3 

2.1 

1.8 

1.6 

21 

.150 

4200 

30.4 

5.9 

5.1 

4.6 

3.9 

3.5 

3.1 

2.9 

2.5 

2.3 

3 

.193 

6900 

36.4 

7.3 

6.4 

5.8 

5.0 

4.4 

4.0 

3.7 

3.2 

2.9 

31 

.236 

10300 

42.7 

8.6 

7.6 

6.9 

6.0 

5.3 

4.8 

4.5 

3.9 

3.5 

4 

1 

.258 

12200 

48.6 

9.1 

8.1 

7.4 

6.4 

5.7 

5.2 

4.8 

4.3 

3.8 

41 

1 

.301 

16600 

54.7 

10.3 

9.3 

8.5 

7.4 

6.6 

6.1 

5.6 

4.9 

4.5 

5 

1 

.344 

21700 

60.8 

11.4 

10.3 

9.5 

8.3 

7.5 

6.8 

6.3 

5.6 

5.1 

51 

1 

.387 

27500 

67.0 

12.5 

11.3 

10.5 

9.2 

8.3 

7.6 

7.1 

6.3 

5.7 

6 

11 

.408 

30600 

72.9 

12.8 

11.710.9 

9.6 

8.6 

7.9 

7.4 

6.6 

6.0 

7 

11 

- 

.494 

44800 

85.4 

14.8 

13.6 

12.7 

11.3 

10.2 

9.4 

8.8 

7.8 

7.1 

8 

U 

.580 

61800 

97.9 

16.7 

15.4 

14.5 

12.9 

11.7 

10.9 

10.2 

9.1 

8.3 

9 

11 

.644 

76300 

109.6 

17.9 

16.6 

15.6 

14.0 

12.8 

11.9 

11.1 

10.0 

9.1 

10 

\\ 

f 

.730 

98000 

121.8 

19.5 

18.2 

17.2 

15.5 

14.3 

13.3 

12.5 

11.2 

10.3 

12 

\\ 

1 

.902 

149500 

146.3 

22.6 

21.2 

20.1 

18.4 

17.0 

15.9 

15.0 

13.5 

12.4 

CHAPTER  VII. 

BUILDING  CONSTRUCTION. 

153.  Division   of  the  Subject.— The  various  elements  of 
building   construction    relating   to    reinforced-concrete    design 
may  be  grouped  under  the  following  heads:    (1)  Beams  form- 
ing a  continuous  surface,  as  floor-  and  roof -slabs;    (2)  Floor- 
beams  and  girders;   (3)  Columns;   (4)  Footings;   (5)  Walls  and 
partitions.     In  the  discussion  of  these  various  elements  con- 
sideration will  be  given  to  the  determination  of  stresses,  the 
design  of  the  members,   and  the  arrangement  of  connective 
details. 

154.  General  Arrangement  of  Concrete  Floors. — Two  gen- 
eral types  of  floors  may  be  considered:    (1)  that  in  which  the 
floor- slab  is  supported  on  steel  beams,  and  (2)  that  in  which 
concrete  beams  are  used,  the  floor  of  the  entire  structure  being 
of  a  monolithic  character.     In  the  former  case  the  steel  skele- 
ton consists  of  columns,  girders,  and  cross-beams,  the  beams 
being  spaced    commonly  about  6  feet  apart.    The    floor-slab 
is  supported  mainly  by  the  cross-beams.    The  same  variety 
of  arrangements  is  used  in  the  case  of  all-concrete  structures, 
the  cross-beams  being  spaced  usually  from  four  to  six  feet 
apart.    The  cross-beams  may  in  this  case  be  entirely  omitted, 
giving  span  lengths  of  15  to  20  feet.     Sometimes,  also,  the 
cross-beams  are  inserted  only  at  columns,  forming  a  nearly 
square  panel  of  the  floor-slab,  which  is  then  considered  as  sup- 
ported on  four  sides. 

155.  Stresses  in  Continuous  Beams. — Since  floor-slabs  and 

beams  are  commonly  designed  to  act  as  continuous  beams 

237 


238  BUILDING  CONSTRUCTION.  [Cn.  VIL 

it  is  important  to  investigate  the  possible  stresses  under  such 
conditions,  although  exact  calculation  is  impracticable  and 
unnecessary.  In  the  case  of  floor-slabs  there  is  usually  a 
large  number  of  consecutive  spans  and  the  loading  producing 
the  theoretical  maximum  moments  at  various  points  would 
involve  unreasonable  assumptions  as  to  position  of  live  loads. 
Sufficiently  exact  analysis  may  be  arrived  at  by  considering 
certain  simple  cases.  Take,  for  example,  the  two  cases  of  a 
beam  of  two  equal  spans  and  one  of  three  equal  spans,  in  each 
case  the  beam  being  " supported"  at  the  ends.  Assume  both 
a  fixed,  or  dead-load,  and  a  moving,  or  live-load,  and  that 
the  latter  consists  of  a  uniform  load  distributed  over  such 
portion  of  the  floor  as  to  cause  the  maximum  moment  at  each 
section.  This  maximum  moment  is  readily  calculated  by 
means  of  the  usual  continuous  girder  formulas.  The  calculation 
will  not  be  repeated  here.  The  results  are  shown  graphically  to 
scale  in  Figs.  71  and  72.  The  dotted  lines  relate  to  dead- load 
effects  and  the  dashed  lines  to  live-load  effects.  The  span 
length  and  the  load  per  foot  are  assumed  equal  to  unity.  If, 
in  any  given  case,  w=dead  load  per  foot,  p  =  live  load  per  foot, 
and  Z=span  length,  the  true  moments  will  be  found  by  mul- 
tiplying the  proper  ordinates  by  wl2  or  by  pi2  respectively. 

In  Fig.  71  the  maximum  positive  moment  =  .Q70wl2  +  .095pZ2, 
and  the  maximum  negative  moment  =  JwZ2  +  JpZ2. 

In  Fig.  72  the  maximum  positive  moments  are: 

First  span,      M  =  .OSOwl2  +  .lOOp/2. 
Second  span,  M  =  .Q25wl2+  ,Q75pl2. 

The  maximum  negative  moments  at  supports  are  each 
M  =  .mwl2+.117pl2. 

Assuming,   for   example,    that   the  dead   load   is    one-half 
the  live  load,  then  there  results  for  the  first  case 

Maximum  positive  moment  --=. 087 (w+p)l2, 
Maximum  negative  moment  =  J(w+  p)l2, 


§  155.1  STRESSES   IN   CONTINUOUS  BEAMS. 

and  for  the  second  case 

Maximum  positive  moment  =  . 093 (w+p)l2, 
Maximum  negative  moment  =  .111  (w+  p)l2. 


239 


0.2 


0.1 


0.2 


0.1 


0.0 


0.1 


0.2 


0.1 


0.2 


\l 


Dead  Load 

Liye-Load 

Total  Load 


FIG.  71. — Moments  in  Beams  of  Two  Spans. 


0.1 

0  1 

s 

^ 

s 

x 

/ 

\ 

/ 

\ 

/ 

/ 

'' 

"^ 

\ 

s~ 

•^^ 

/ 

^ 

s* 

•\ 

s 

s 

0.0 

/  / 

's' 

^^ 

S\ 

S- 

N, 

/; 

.'' 

^N\ 

A 

/ 

f 

y^ 

A, 

/ 

' 

s 

^ 

/ 

^ 

^ 

I* 

\ 

V. 

/ 

t 

^ 

4 

/ 

A 

£ 

***'*^ 

**~~*^ 

>>, 

\ 

\ 

•-N, 

// 

/ 

^\ 

\x 

s 

/ 

/ 

s* 

-- 

-~«N 

^ 

0.1 
n^ 

~~ 

-•  — 

—  — 

X 

\ 

\ 

/ 

/ 

\ 

\ 

/ 

/ 

/ 

,__- 

' 

* 

—  •-. 

•—  _ 

.s 

\ 

s 

^ 

/ 

N 

^, 

/ 

i 

v. 

_.- 

,  —  -- 

v 

. 

/ 

-^f 

— 

±s 

Xr 

3 

f 

4 

\ 

5 

// 

/ 

\s 

\\ 

// 

/ 

L\\ 

/! 

/ 

\ 

\\ 

/// 

\^ 

1  I 

\,  N 

// 

0  ° 

\ 

1 

\ 

/ 

\ 

I 

5 

/ 

).ead  Loaii 
rfvft  Load 


FIG.  72.—  Moments  in  Beams  of  I'hree  Spans. 

In  Figs.  71  and  72  the  full  lines  represent  the  maximum 
moments  throughout  the  beam  for  the  condition  that  w  =  \p\ 


240  BUILDING  CONSTRUCTION.  [Cn.  VII. 

these  lines  are  particularly  useful  in  showing  the  relative  dis- 
tances from  the  supports  over  which  positive  and  negative 
moments  may  occur. 

Similar  calculations  might  be  made  for  a  greater  number 
of  spans,  but  the  results  for  the  second  case  may  be  taken  as 
approximately  what  would  be  found  for  a  larger  number.  We 
may  conclude,  therefore,  that  for  three  or  more  spans  the  maxi- 
mum moment  is  approximately  tir  (w+p)l2.  This  is  in  accord- 
ance with  a  common  rule  of  practice.  The  maximum  shears 
near  supports  are  not  greatly  affected  by  moving  loads.  For 
intermediate  spans  the  maximum  end  shear  may  be  taken  at 
one-half  of  the  span  load;  for  end  spans  the  shear  near  the 
second  support  will  be  approximately  six-tenths  of  a  span 
load. 

These  suggested  values  of  moments  should  be  modified 
where  the  relation  of  live  to  dead  load  is  greatly  different 
from  that  here  assumed.  Where,  for  example,  the  load  is  all 
fixed  load,  the  center  moments  would  be  much  smaller,  and  in 
the  case  of  three  or  more  spans  the  moments  over  the  sup- 
ports would  be  somewhat  smaller  than  here  estimated. 

156.  Effect  of  Rigid  Supports  on  the  Resisting  Moment. 
If  a  flat  slab  is  held  between  unyielding  supports,  such  as  fixed 
I-beams,  a  strength,  or  resisting  moment,  will  be  developed 
in  the  slab  even  though  there  be  no  steel  reinforcement.  Fail- 
ure cannot  take  place  without  the  crushing  of  the  concrete 
either  at  the  center  or  at  the  support.  For  short  spans  this 
resisting  moment  (the  so-called  "arch  action")  is  about  as 
great  as  will  exist  in  the  slab  if  reinforced  and  simply  sup- 
ported at  the  ends.  In  the  case  of  a  flat  reinforced  slab  such 
rigid  supports  likewise  add  considerably  to  the  strength  of 
the  slab,  giving  the  effect  of  partial  continuity.  This  strengthen- 
ing effect  of  rigid  supports  is  roughly  proportional  to  the  square 
of  the  slab  thickness  and  inversely  proportional  to  the  square 
of  its  length. 

In  practice,  the  supports  of  slabs  of  short  span  length, 
whether  consisting  of  I-beams  or  of  concrete  beams  of  which 


§  158.]  SQUARE   SLABS.  241 

the  slab  is  a  part,  are  rendered  very  rigid  by  reason  of  the 
action  of  the  adjoining  floor-panels.  Even  where  the  slabs 
are  simply  supported  on  the  tops  of  steel  beams  the  adjoining 
slabs  prevent  to  some  extent  lateral  motion,  rendering  all  such 
spans  partially  continuous.  The  strengthening  effect  of  rigid 
supports  is,  therefore,  especially  great  in  the  case  of  narrow 
floor-spans  and  where  there  is  a  large  number  of  consecutive 
unbroken  panels.  Under  such  conditions  reinforcement  against 
negative  moment  is  hardly  necessary.  For  long  spans  and  for 
spans  on  the  outside  of  a  system  the  effect  is  small. 

157.  Slabs  Reinforced  in  Two  Directions. — If   the   panel 
between  beams  is  square,  or  nearly  so,  the  slab  may  advan- 
tageously be  reinforced  in  both  directions.     The  exact  analysis 
of  stresses  in  such  a  case  is  difficult,  if  not  impossible,  as  the 
effect  of  the  more  or  less  rigid  supports  is  especially  important 
and  the  problem  is  otherwise  difficult  of  exact  treatment. 

The  following  solution  for  square  and  rectangular  slabs  will 
serve  to  show,  approximately,  the  relation  of  the  loads  car- 
ried by  the  two  systems  of  reinforcement.  The  results  are 
certainly  safe  and  do  not  vary  much  from  rules  of  practice, 
but  point  to  a  somewhat  more  economical  use  of  material. 

158.  Square  Slabs. — In  this  case  the  reinforcement  should 
be  of  equal  amount  in  the  two  directions.     It  may  be  calcu- 
lated on  the  assumption  that  one  half  the  load  is  carried  by 
each  system  of  reinforcement.     The  concrete  is  proportioned 
for  only  one  system,  or  one-half  the  load,  as  the  stresses  due 
to  the  two  systems  are  at  right  angles  to  each  other  and  it  is 
assumed  that  the  stresses  in  one  direction  do  not  weaken  the 
concrete  with  respect  to  stresses  in  the  other  direction.     The 
loading  on  each  system  is  usually  assumed  to  be  uniformly 
distributed,  resulting  in  an  equal  spacing  of  rods  throughout 
the  beam.     This  assumption  is,  however,  far  from  the  truth, 
and  while  giving  safe  results  it  is  desirable  to  consider  a  more 
exact  analysis  of  the  problem  which  will  show  that  the  rods 
should  be  spaced  closer  at  the  center  than  at  the  edge. 

In  Fig    73,  A  BCD  represents  a  square  slab  supported  on 


242 


BUILDING  CONSTRUCTION. 


[On.  VII. 


all  sides  and  loaded  with  a  uniform  load  w  per  unit  area. 
Consider  the  relative  amounts  of  load  carried  by  the  system 
parallel  to  aa'  and  the  system  parallel  to  mm! .  At  the  centre 
0,  and  at  all  points  on  the  diagonal  lines  AD  and  CB,  it  fol- 
lows from  symmetry  that  the  loading  is  equally  distributed  on 
the  two  systems  and  is  equal  to  w/2.  At  point  E  the  pro- 
portion of  the  load  carried  by  the  system  aa'  will  be  much 
greater  than  that  carried  by  the  system  mm',  since  for  given 
loads  the  beam  element  along  aa'  will  deflect  much  less  at 
point  E  than  will  the  element  along  mm' .  In  general,  there- 
fore, as  we  approach  the  support  BD  the  proportion  of  load 
carried  by  the  system  aa'  increases,  reaching  a  value  of  w  at 

A  in B 


V 

/f 
y/ 

1 

~"T"" 

I 

li 

A 

K 

7                               v 

B 

FIG.  74. 

FIG.  73. 

the  extreme  end  a'.  The  distribution  of  load  on  aa'  may  then 
be  roughly  represented  by  the  ordinates  from  AB  to  the  curved 
line  aOa'  of  Fig.  74.  Consider  now  the  load  along  a  line  66'. 
At  points  F  and  F'  the  load  will  be  w/2;  at  point  G'  it  will 
be  less  than  w/2,  being  the  same  as  the  load  on  the  system  mmf 
at  G.  It  will  be  shown  in  Fig.  74  by  the  ordinate  GH  from 
Oaf  to  the  line  aa'.  At  the  end  6'  the  load  will  be  w.  The 
curve  of  distribution  will  then  be  somewhat  as  represented 
by  the  line  off  a'  in  Fig.  74,  in  which  G'K=GH. 

Assuming  the  curve  aOa'  to  be  a  parabola  it  is  found  that  the 
centre  bending  moment  along  the  line  aa',  for  a  beam  one 
foot  wide,  will  be  ^  (w/2)l2  instead  of  ^  (w/2)l2,  as  results 


§  159.]  RECTANGULAR  SLABS.  243 

from  the  usual  assumption.  The  spacing  of  the  rods  at  the 
centre  may  then  be  determined  on  this  basis.  At  points  inter- 
mediate between  the  centre  and  the  edge,  the  rods  might  well 
be  spaced  so  that  the  number  per  foot  would  vary  from  the 
required  number  at  the  centre  to  zero  at  the  edge,  following 
the  law  of  the  parabola.  If  N  represents  the  total  number 
required  on  the  ordinary  assumption  of  equal  spacing,  then 
|A7X|,  or  %N,  would  represent  the  more  correct  number 
when  spaced  as  here  calculated.  Practically  as  good  results 
will  be  secured  if.  the  rods  are  spaced  uniformly  at  the  usual 
spacing,  determined  by  the  formula  M  =  %(w/2)l2,  for  the  cen- 
tre half  of  the  slab,  then  gradually  reduce  the  number  per 
foot  to  the  edge  of  the  slab,  using  one-half  as  many  rods  for 
the  remaining  two  quarters.  The  total  number  used  would 
then  be  J2V  instead  of  JAT  as  above  determined,  but  the 
strength  would  be  ample.  If  the  slabs  are  continuous,  then 
TV  or  ^  should  be  substituted  for  J  in  the  formula  for  M,  as 
may  be  permissible. 

159.  Rectangular  Slabs  of  Greater  Length  than  Breadth. — 
As  a  slab  becomes  oblong  in  form  the  relative  amount  of  load 

carried  by  the  longitudinal  system     A m B 

becomes  rapidly  less.  Consider  the 
case  of  a  slab  twice  as  long  as 
wide  (Fig.  75).  For  equal  fibre 
stresses  the  longitudinal  system  aaf 
will  deflect  four  times  as  much  as  c  D 

the  transverse  system  mm'.    Hence  FlG>  75- 

the  same  deflection  involves  only  one  fourth  the  unit  stress  on 
the  bars  aa'  as  on  the  bars  mm'.  If  equal  spacing  be  used,  then 
the  load  carried  by  the  longitudinal  system  will  be  that  which  pro- 
duces one-fourth  the  fibre  stress  as  in  the  system  mm! .  Finally, 
since  the  stress  for  given  loads  is  proportional  to  the  span,  we 
find  that  only  one-eighth  as  much  load  will  be  carried  by  the  lon- 
gitudinal system  as  by  the  transverse  system  with  equal  spacing 
of  rods.  For  points  nearer  the  ends  of  the  slab  the  proportion 
carried  by  the  longitudinal  system  will  be  greater,  but  in  any 


244  BUILDING  CONSTRUCTION.  [Cn.  VII. 

case  the  longitudinal  rods  will  be  much  under-stressed.  If 
the  length  is  1.25  times  the  breadth,  then  the  working  stress  in 
the  longitudinal  rods  will  be  about  two-thirds  that  in  the 
cross-rods  (at  the  centre)  and  they  will  carry  about  one-half 
as  much  load  when  spaced  the  same. 

From  this  discussion  it  is  evident  that  longitudinal  rein- 
forcement should  not  be  used  to  carry  load  in  oblong  panels 
where  the  length  exceeds  the  breadth  by  more  than  15  to  20  %. 
An  excess  of  25%  would  seem  to  be  about  the  practical  limit. 
Whatever  steel  is  placed  in  the  longitudinal  direction  is  used 
uneconomically. 

1 60.  Reinforcement  to  Prevent  Cracks. — While  longitudinal 
reinforcement  is  of  little  value  in  carrying  loads,  a  small  amount 
is  nevertheless  often  desirable   in    preventing   cracks  and  in 
binding   the   entire   structure  together.       For    a    close  beam 
spacing  such  reinforcement  is  hardly  necessary,  as  the  beam 
reinforcement  itself  thoroughly  ties  the  structure  longitudinally 
along  the  beam  lines.     For  wide  beam  spacing  it  is  more  im- 
portant.    Just  what  amount  of  steel  is  needed  is  a  matter  of 
experience.    The  use  of  J-inch  or  f-inch  rods  spaced   about 
two  feet  apart  is  common  practice.     If  a  metal  fabric  is  used 
for  oblong  panels,  the  longitudinal  metal  should   be  propor- 
tioned in  accordance  with  the  principles  discussed  in  this  and 
the  preceding  articles. 

161.  Floor-slabs  Supported  on  Steel  Beams.— Many  "sys- 
tems" have  been  developed  of  this  type  of  construction,  differ- 
ing from  each  other  in  form  of  steel  used,  position  of  the  con- 
crete relative  to  the  beam,  use  of  curved  or  flat  slabs,  use  of 
various  kinds  of  hollow  tile  in  connection  with  the  concrete, 
etc.      Sufficient    examples    only  will    be  given  to  illustrate 
the  principles  involved ;  further  information  regarding  the  many 
systems  can  readily  be  had  from  trade  catalogues. 

Fig.  76  shows  the  floor  placed  directly  on  the  tops  of  the 
beams.  The  reinforcement  may  be  small  rods  or  a  mesh- 
work  of  expanded  metal  or  woven  fabric.  If  reinforced  as 
shown,  the  slab  must  be  calculated  as  a  simple  beam,  there 


§  161.] 


RECTANGULAR  SLABS. 


245 


being  no  reinforcement  against  negative  moment  over  the  sup- 
port. For  spans  of  considerable  length  some  reinforcement 
for  negative  moments  is  desirable  to  secure  economy  and  to 


FIG.  76. 


prevent  cracks  in  the  upper  surface,  although  the  lateral 
rigidity  due  to  adjoining  panels  is  of  much  assistance,  as- 
explained  in  Art.  156. 

Fig.  77  represents  a    slab  constructed  after  Hennebique's- 


FIG.  77. 

system,  to  be  supported,  by  walls  or  steel  beams.  Small  rods 
are  used  for  reinforcement,  every  alternate  rod  being  bent  up 
and  stirrups  of  flat  steel  looped  on  the  straight  rods.  This 
is  a  very  effective  design  to  secure  strength  against  shear  or 
diagonal  tensile  stresses,  but  except  where  the  floor-load  is 
very  heavy  special  shear  reinforcement  is  hardly  needed  in 
floor-slabs.  Fig.  78  shows  a  more  common  design  of  non- 


FIG.  78. 


continuous  slab,  the    concrete  being  supported  on  the  lower 
flange  and  the  entire  beam  surrounded. 

Fig.  79  shows  a  standard  form  of  construction   in  which 


FlG.  79. 


the  slab  is  practically  continuous.     The  reinforcing  material 
may  be  rods  or  a  metal  fabric  continuous  over  several  spans. 


246 


BUILDING  CONSTRUCTION. 


[CH.  VII. 


Figs.  80  and  81  show  two  forms  in  which  a  bar  is  hooked 
around  the  beam  flange. 


FIG.  80. 


FIG,  81. 


Many  other  forms  are  employed,  some  using  a  concrete  arch 
with  more  or  less  reinforcement.  In  some,  also,  the  concrete 
slab  is  brought  down  somewhat  below  the  beam,  giving  a  plane 
surface  on  the  under  side. 

162.  Floor-slabs  in  All-concrete  Construction.— Where  con- 
crete beams  are  used  the  slab  and  beam  are  usually  built  simul- 
taneously, giving  a  monolithic  structure.  The  slab  thus  con- 
stitutes part  of  the  beam,  but  to  be  effective  these  two  parts 
must  be  well  tied  together.  Where  cross-beams  are  used  the 
span  of  the  slab  will  commonly  range  from  4  to  6  feet  in  length. 
For  such  short  spans  a  reinforcement  of  rods  or  metal  mesh- 
work  near  the  bottom  only  will  be  effective  as  explained  in  Art. 
156.  This  reinforcement,  if  of  rods,  should  be  laid  with  lapped 
and  broken  joints  to  give  continuity  and  to  prevent  the  localiza- 
tion of  contraction  cracks  in  undesirable  places  (Fig.  82).  The 


____ 


FIG.  82. 

beam,  if  well  bonded  to  the  slab,  will  make  a  very  rigid  sup- 
port comparable  to  the  I-beam. 

In  the  case  of  spans  longer  than  5  to  6  feet  it  becomes  desir- 
able to  reinforce  against  negative  moment.  This  may  readily 
be  done  by  bending  up  a  part  or  all  of  the  rods  and  extending 


§  162.] 


FLOOR-SLABS. 


247 


the  bent  ends  beyond  the  beam.    Fig.  83  illustrates  two  arrange- 
ments of  this  kind.     In  either  case  the  amount  of  steel  at  the 


4 


FIG.  83. 


top  above  the  beam  is  the  same  as  at  the  bottom  in  the  centre 
of  the  slab.  The  result  may  also  be  arrived  at  by  using  sep- 
arate straight  rods,  as  shown  in  Fig.  84.  The  plan  of  bent 


FIG.  84. 

rods  has  a  slight  advantage  as  it  reinforces  somewhat  against 
shearing  failures,  but  this  is  not  usually  important  in  slabs. 
For  very  heavy  loads,  however,  it  becomes  of  importance,  and 
the  same  care  should  be  used  as  in  the  design  of  large  beams. 
Fig.  85  shows  the  Hennebique  bent-rod  and  stirrup  sys- 
tem applied  to  long-span  slabs. 


FIG.  85 

The  length  of  span  over  which  negative  moment  is  likely 
to  exist  may  be  estimated  from  Fig  72.  It  is  seen  that  in 
the  centre  span  of  a  three-span  girder,  where  the  dead  load  is 
one- half  the  live  load,  negative  moment  may,  under  extreme 
conditions,  occur  entirely  across  the  beam.  For  long  spans  a 
top  reinforcement  at  least  to  the  third  point  will  be  desirable, 


248  BUILDING  CONSTRUCTION.  [Cn.  VII. 

but  for  short  spans  a  less  extensive  reinforcement  will  be 
sufficient.  The  effect  of  a  less  amount  of  steel  is  discussed 
in  Art.  165.  Other  examples  of  slab  construction  are  shown 
in  Art.  168. 

163.  Beams  and  Girders. — Economical  Arrangement.— The 
arrangement  of  columns,  girders,  and  beams  is  determined 
according  to  the  same  principles  as  in  steel  construction.  The 
spacing  of  columns  and  girders  will  be  determined  largely  by 
architectural  considerations.  The  best  spacing  of  cross-beams 
will  differ  in  different  cases.  Where  the  spacing  of  girders  is 
not  too  great  (12  to  15  feet)  and  where  cross-beams  are  not 
needed  to  secure  lateral  stiffness,  it  will  be  a  question  of  omit- 
ting all  cross-beams,  of  inserting  them  only  at  columns  so  as 
to  form  a  square  or  nearly  square  panel,  or  of  spacing  them 
at  closer  intervals  of  4  to  8  feet,  using  two  or  more  to  a  girder- 
panel.  The  preceding  analysis  shows  that  double  reinforce- 
ment will  not  be  economical  for  oblong  panels.  Cross-beams, 
if  used,  should  therefore  be  arranged  to  give  very  nearly  square 
panels  or  else  be  spaced  much  more  closely,  designing  the  rein- 
forcement so  as  to  carry  the  entire  load  to  the  beams  and 
thence  to  the  girders. 

If  not  otherwise  needed,  the  use  of  cross-beams  to  secure 
square  panels  effects  little  if  any  saving.  The  amount  of  con- 
crete will  be  less,  but  the  amount  of  steel  required  will  be  more, 
and  the  extra  beam  will  be  more  costly  per  unit  volume  than 
the  slab.  However,  for  the  sake  of  lateral  stiffness  it  will 
usually  be  desirable  to  place  cross-beams  at  columns. 

Where  close  spacing  of  beams  is  adopted  the  best  arrange- 
ment depends  upon  the  loading  and  the  working  stresses,  as 
well  as  upon  the  cost  of  the  material  and  forms.  Heavy  loads 
and  low  stresses  call  for  large  weights  of  concrete  and  tend  to 
require  the  use  of  the  material  more  in  the  form  of  deep  ribs 
or  beams,  as  the  deeper  the  beam  the  greater  its  moment  of 
resistance  for  a  given  volume.  If  cross-beams  are  used,  a 
spacing  greater  than  10  or  12  feet  or  less  than  4  or  5  feet  will 
seldom  be  economical.  Architectural  considerations  will  often 


§  164.] 


BEAMS  AND   GIRDERS. 


249 


govern,  and  frequently  building  regulations  relative  to  ratio 
of  span  to  depth  will  control. 

164.  Distribution  of  Floor-loads  to  Beams. — Where  the  floor- 
slab  is  reinforced  in  one  direction  only  the  load  will  prac- 
tically all  be  transmitted  to  the  corresponding  beams,  but  at 
the  ends  of  the  panels  a  small  part  will  be  transferred  directly 
to  the  girder.  This  may  be  neglected  in  the  calculations.  In 
the  case  of  reinforcement  in  two  directions,  unless  the  panel 
is  nearly  square,  the  load  may  still  be  assumed  as  all  trans- 
ferred to  the  side  beams.  If  the  panels  are  square,  or  nearly 
so,  the  distribution  may  be  assumed  in  accordance  with  the 
discussion  of  Art.  158.  Thus  the  load  brought  to  point  a' 
(Fig.  74)  will  be  one-half  of  the  area  below  the  curve  aOa', 


FIG.  86. 

and  the  load  brought  to  b'  will  be  one-half  the  area  below  the 
curve  bG'b',  etc.  The  distribution  along  the  beam  will  then 
follow  some  such  law  as  represented  by  the  shaded  area  in 
Fig.  86  (a),  the  total  load  being  necessarily  \wl,  where  w  =  floor- 
load  per  square  foot.  It  will  be  sufficiently  accurate  to  assume 
this  curve  a  parabola.  The  centre  bending  moment  in  the 
beam,  assumed  as  a  simple  beam,  will  then  be  equal  to 


A  distribution  of  load  as  represented  in  Fig.  86  (6),  as  is  some- 
times assumed,  gives  a  centre  moment  equal  to  -^ui2,  a  value 
about  7%  higher  than  the  above.  A  uniform  distribution  gives 
a  moment  equal  to  §^u72,  a  value  20%  lower. 


250  BUILDING  CONSTRUCTION.  [Cn.  VII. 

165.  Design  of  Cross-beams. — In  the  design  of  beams  the 
chief  features  are  the  determination  of  the  cross-section,  the 
amount  of  steel  and  its  make-up,  provision  for  shearing  stress, 
provision  for  negative  bending  moment  and  connections  with 
slabs,  other  beams,  and  columns.  The  proportions  of  the  beam, 
whether  considered  as  a  rectangular  beam  or  as  a  T-beam, 
will  be  determined  by  considerations  discussed  in  Chapter  V. 
Ratios  of  depth  to  width  greater  than  2  or  2|  are  seldom  used. 
Requirements  of  head-room,  space  for  rods,  and  shearing 
strength  will  limit  the  possible  variations  in  proportions  to  a 
comparatively  narrow  range.  Deep  beams  are  economical  of 
concrete  but  cost  more  for  forms  than  do  shallow  beams. 

If  the  beam  may  be  calculated  as  a  T-beam,  the  width  of 
slab  which  may  be  counted  on  as  a  part  of  the  beam  is  an  im- 
portant question.  Specifications  usually  allow  a  width  of  six 
to  ten  times  the  thickness  of  the  slab,  but  not  to  exceed  the 
width  between  beams.  As  regards  strength  it  would  be  very 
difficult  to  secure  so  thorough  a  reinforcement  of  web  as  to 
make  it  possible  to  crush  a  flange  as  much  as  four  times  the 
width  of  the  web;  the  excessive  shearing  stresses  in  the  wreb 
would  cause  failure.  As  regards  stiffness,  which  controls  the 
position  of  the  neutral  axis,  the  width  of  the  slab  to  be  counted 
as  part  of  the  beam  may  and  should  be  taken  relatively  great. 
The  width  of  flange  being  known,  the  design  of  the  T-beam 
consists  chiefly  in  the  design  of  the  web  and  the  calculation 
of  the  steel  cross-section.  It  will  be  only  in  the  case  of  large 
girders  that  the  compressive  stress  in  the  concrete  will  be  a 
determining  factor.  Usually  there  is  a  large  excess  of  material. 

If  the  beam  is  to  be  considered  as  continuous  over  sup- 
ports, the  moment  of  resistance  at  the  support  must  also  be 
investigated.  At  this  point  the  tension  side  is  uppermost  and 
the  effective  beam  is  now  a  rectangular  beam.  The  maximum 
moment  is  about  the  same  as  at  the  centre,  thus  requiring 
about  the  same  amount  of  steel  at  the  top  as  is  required  in 
the  centre  of  the  span  at  the  bottom.  The  maximum  com- 
pression in  the  concrete  will  be  greater  than  in  the  centre  and 


§  165.]  BEAMS  AND  GIRDERS.  251 

will  probably  determine  the  size  of  beam  required  unless,  as  is 
often  done,  the  depth  of  beam  is  increased  near  the  end.  (See 
Fig.  87.)  Furthermore,  if  a  part  of  the  bottom  steel  is  carried 


FIG.  87. 

well  through  the  joint,  it  furnishes  considerable  compressive 
reinforcement  at  this  point.  The  necessary  top  steel  at  the 
end  may  be  provided,  as  in  the  slab,  by  bending  up  a  portion 
of  the  lower  rods,  or  by  using  separate  short  rods,  or  by  both 
methods  combined.  To  provide  thoroughly  for  negative 
moment  the  upper  reinforcement  should  extend  to  about  the 
third  point,  and  in  some  cases  still  farther.  Various  arrange- 
ments of  bent-up  rods  are  illustrated  in  the  examples  cited  in 
Art.  168. 

It  is  often  required  that  beams  shall  be  calculated  as  simple 
beams,  using  JwZ2  for  the  maximum  moment.  In  such  a  case 
it  is  still  desirable  to  provide  some  steel  at  the  top  over  sup- 
ports to  prevent  cracks.  The  beams  will  then  act  as  partially 
continuous,  the  flexibility  over  the  support  being  greater  than 
when  fully  reinforced.  This  results  in  some  excess  of  stress  in 
this  steel,  but  without  danger.  The  presence  of  this  steel  should 
be  taken  account  of  in  arranging  the  web  reinforcement. 

The  treatment  of  girders  is  the  same  as  described  for  beams, 
it  being  especially  important  that  the  reinforcement  pass  well 
through  the  column. 

The  arrangement  of  shear  or  web  reinforcement  for  beams 
and  girders  is  of  great  importance,  as  it  is  in  these  forms  where 
the  web  tensile  stresses  will  be  high.  At  points  wrhere  the 
allowable  shearing  stress  in  the  concrete  is  exceeded  steel  must 
be  added  in  some  form  to  carry  a  part  of  the  stress,  as  explained 
in  Art.  125.  Where  bent-up  rods  are  used,  as  in  Fig.  87,  these 
rods  aid  greatly  in  carrying  shear,  and  where  not  spaced  too 
widely  may  be  counted  on  to  add  perhaps  50%  to  the  strength 


252  BUILDING  CONSTRUCTION.  [Cn.  VII. 

of  the  web.  For  thorough  web  reinforcement  the  stirrup  is 
usually  employed,  or  some  form  of  bent  bar  closely  spaced. 
This  reinforcement  may  be  calculated  as  explained  in  Art.  125,  not 
too  much  reliance  being  placed  on  one  or  two  bent  rods.  Web 
reinforcement  will  usually  be  needed  only  for  the  end  quarter 
or  third  of  the  beam.  Near  the  support,  where  the  moment  is 
negative,  the  tendency  is  for  diagonal  cracks  to  start  at  the 
top,  while  farther  along  the  cracks  tend  to  start  at  the  bottom, 
as  shown  in  Fig.  87.  Stirrups  at  points  of  negative  moment 
should  loop  about  the  upper  bars,  and  at  points  of  positive 
moment  should  loop  about  the  lower  bars.  A  correct  appre- 
ciation of  the  diagonal  stresses  in  such  continuous  beams  is 
important. 

The  beam  should  be  well  bonded  to  the  slab,  especially 
near  the  end  where  the  differential  stresses  between  the  two 
parts  are  large.  This  is  well  accomplished  by  means  of  the 
bent  rods  brought  up  as  high  as  possible,  and  by  means  of  the 
slab  reinforcement  which  crosses  the  beam.  Along  the  centre 
of  the  beam  the  matter  is  not  of  so  great  importance,  but  it 
is  better  to  provide  such  bond  by  some  form  of  vertical  rein- 


(c) 


forcement,  such  as  stirrups,  extending  up  into  the  slab  at 
occasional  intervals.  This  is  of  especial  importance  in  the  case 
of  girders  where  the  main  slab  reinforcement  runs  parallel  to 
the  beam.  A  good  bond  is  also  more  necessary  the  thinner  the 
sections.  Sections  shown  in  Fig.  88,  (a)  and  (6),  are  more 
favorable  than  such  a  section  as  in  Fig.  88  (c).  Sharp  reentrant 
angles  in  such  a  brittle  material  as  concrete  are  points  of  weak- 
ness, and  where  they  exist  a  steel  bond  is  desirable. 

1 66.  Columns. — There  is  little  to  be  said  here  relative  to 
column  design.     Much  difference  of  opinion  still  exists  as  to 


§  167.]  COLUMNS.  253 

the  use  of  large  or  small  quantities  of  steel  and  methods  of 
calculation.  A  conservative  course  should  be  pursued  in  this 
matter,  as  the  columns  and  beams  in  a  reinforced  structure 
are  the  vital  parts  of  the  structure.  Working  stresses  in 
columns  such  as  700  or  800  lbs/in2  should  not  be  employed. 
Where  large  areas  of  steel  are  used,  and  figured  at  ordinary 
working  stresses,  such  steel  skeleton  should  not  rely  upon 
the  concrete  for  rigidity.  Concrete  may,  however,  be  relied 
upon  to  transmit  loads  from  girders  to  columns.  Where  small 
areas  of  steel  are  used  the  rods  should  be  well  lapped  at  the 
floor-level,  and  those  from  the  lower  columns  should  extend 
upwards  the  full  depth  of  the  connecting  beams.  The  rods 
should  be  well  banded  together  by  steel  bands  or  large  wire 
so  as  to  hold  all  parts  in  position  and  to  strengthen  the  column 
circumferentially.  Unless  such  banding  is  spaced  very  closely 
it  should  not  be  counted  upon,  however,  as  " hooping." 
Brackets  under  all  connecting  girders  are  serviceable  in 
stiffening  the  frame  as  well  as  in  decreasing  the  stress  in  the 
girders.  Rods  of  connecting  girders  should  pass  well  through 
the  columns. 

167.  Eccentric  Loads  on  Columns. — Where  loads  are  applied 
on  free  brackets  or  cantilevers  the  load  is  definitely  eccentric, 
and  the  moment  due  to  the  same,  can  readily  be  calculated. 
Moments  are  also  caused  in  columns  by  unevenly  loaded  panels 
through  the  rigid  beam  connections.  Assuming  the  beams 
rigidly  fixed  at  the  ends,  a  panel  load  on  one  without  a  load 
on  the  corresponding  one  on  the  opposite  side  will  cause  a 
bending  moment  in  the  beam  at  the  column  equal  to  -hpl2, 
where  p  =  live  load  per  lineal  foot  of  beam.  This  moment  is 
resisted  mainly  by  the  column  and  the  members  attached  to 
it  in  the  same  plane  as  the  loaded  beam,  and  in  proportion  to 
their  moments  of  inertia  divided  by  their  lengths.*  If  the  two 
beams  are  about  as  rigid  as  the  column,  then  the  moment  in 
the  column  above  and  below  the  floor  will  be  about  one-fourth 
of  the  given  moment,  =--??pl2.  This  indicates,  roughly,  what 

*  See  Johnson's  Framed  Structures,  Art.  154. 


254 


BUILDING  CONSTRUCTION. 


[On.  VII. 


may  be  expected  from  unequally  loaded  floors.  In  the  lower 
stories  of  a  high  building  such  a  moment  would  be  of  little 
consequence,  but  in  the  upper  floors  it  might  add  a  large  per- 
centage to  the  column  stress. 

1 68.  Examples  of  Floor  and  Column  Design. — The  follow- 
ing examples  have  been  selected  from  published  designs  as 
representing  good  practice  and  as  illustrating  more  or  less 
specifically  various  features  of  design. 


J/16  Bars  8 'apart,  alternate  bars 
bent  upward  as  jshoaai 

SECTION-  BEAMS 


I 


FIG.  89. — Details  of  the  Robert  Gair  Factory,  Brooklyn. 

Fig.  89  illustrates  the  details  of  the  Robert  Gair  factory, 
Brooklyn.*  The  reinforcement  of  the  columns  varies  from 
eight  l-g-in.  round  rods  at  the  base  to  four  f-in.  rods  at  the 
top.  In  the  lower  stories  the  bars  are  threaded  and  connected 
by  sleeves.  The  rods  are  connected  by  hoops  spaced  from 
4  to  10  in.  apart.  The  girders  are  about  16  ft.  apart,  and  the 


*  Eng.  Record,  Vol.  51,  1905,  p.  279. 


§  168.] 


FLOOR  AND  COLUMN  DESIGN. 


255 


beams  about  one-third  of  this  distance.  Features  of  design 
to  be  noted  are  the  brackets  on  the  columns,  bent  rods  and 
stirrups  in  the  beams,  bent  rods  in  the  slabs,  and  longitudinal 
reinforcement  by  the  use  of  A-inch  bars.  The  stirrups  are 
rather  widely  spaced.  The  Ransome  bar  was  used  except  in. 
the  columns. 

Fig.  90  shows  the  details  of  the  Park  Square  Garage,  Bos- 
ton.*   The  slabs  are  reinforced  with  expanded  metal  brought 


g,^^.^ 

i^U-Trr?:^^::^^ 


2  Rods  1  Diam.^  ^-f  Rods  1  Diam. 

23'0"  for  Front  Beams 


23  8J$  fof  Garage  Beams 
BEAM 


FIG.  90.— Details  of  the  Park  Square  Garage,  Boston. 

near  to  the  top  surface  over  the  beams.  They  were  calcu- 
lated as  continuous  girders.  Beams,  and  girders  were  cal- 
culated as  T  sections  and  figured  on  the  basis  of  375  lbs/in2 
compressive  stress,  and  30  lbs/in2  shearing  stress  with  no 
web  reinforcement,  and  100  lbs/in2  with  such  reinforcement. 
The  column  reinforcement  consisted  of  round  rods  from  2|  in. 
to  J  in.  in  diameter.  They  were  banded  by  J  in.  bands.  The 
concrete  used  in  the  columns  was  1:1J:3;  for  the  lower  parts 
of  the  beams,  1 :2J:4;  and  for  the  upper  parts  of  the  beams  and 
for  slabs  1:2J:5.  The  close  spacing  of  the  stirrups  is  note- 
worthy. 


*  Eng.  Record,  Vol.  52,  1905,  p.  373. 


256 


BUILDING  CONSTRUCTION. 


[Cn.  VII. 


Fig.  91  shows  the  construction  for  the  Thompson  and  Mor- 
ris factory,  Brooklyn.     Corrugated  bars  are  used  throughout. 


,>$  Corr.  Bars  18  Centers 
,  ^"Corr.  Bars  7"Centers 


X'Corr 

-±L-=- 


BEAM  DETAILS 


Columi 

3-%" Corr.  Bars 
Bars-^  ^Corr^  Bars          TV'Long--^ 


7.1fc"Corr.  BarB./X  Columi 

3  of  which  tend  up 

GIRDER  DETAILS 

FIG.  91. — Details  of  the  Thompson  and  Morris  Factory,  Brooklyn. 

The  girders  are  spaced  12  to  15  ft.  apart  and  the  beams  3  ft. 
9  in.  apart,  four  to  a  panel.  The  heavy  reinforcement  for 
negative  moment  should  be  noted. 

Fig.  92  shows  details  of  the  Citizens'  National  Bank  Build- 
ing, Los  Angeles,  Cal.*  Large  girders  connect  the  columns  in 
both  directions,  forming  panels  17  ft.  by  22  ft.  These  panels 
are  then  subdivided  into  four  smaller  ones  by  cross-beams  in 
both  directions,  a  somewhat  peculiar  arrangement  used  prob 
ably  for  architectural  effect.  The  slabs  are  reinforced  both 


Fi©«  92. — Details  of  the  Citizens'  National  Bank  Building,  Los  Angeles. 

ways  by  f-in.  twisted  bars,  4J  in.  apart.     The  stirrups  are  flat 
bands  Tt"X2"  and  spaced  18  ins.  apart  except  near  the  end, 
as  shown.    They  are  looped  about  the  rods  in  a  very  effective 
*  Eng.  News,  Vol.  56,  1906,  p.  16 


§  168.1 


FLOOR  AND  COLUMN  DESIGN. 


257 


manner.     Note  the  sleeve-splice  for  the   column  bars.     The 
beams  and  columns  are  made  of  1:2:3  concrete. 

Fig.  93  shows  a  typical  girder  constructed  with  the  Kahn 
bar  illustrated  in  Fig.  7,  Art.  33.  By  using  inverted  bars  over 
the  supports  negative  moment  can  be  provided  for,  and  at  the 
same  time  additional  shear  reinforcement. 

n n 


FIG.  93. — Reinforcement  with  the  Kahn  Bar. 

In  executing  work  a  practical  difficulty  of  considerable 
importance  is  that  of  placing  -and  keeping  all  bars  in  their 
proper  position  until  the  concrete  is  in  place.  Very  consider- 
able labor  is  required  in  wiring  bars  in  position,  or  in  provid- 
ing other  means  of  support,  and  careful  supervision  is  necessary 
during  construction  to  see  that  they  remain  in  place.  To 
avoid  these  difficulties  various  arrangements  have  been  devised 
for  fastening  together  all,  or  a  part,  of  the  rods  of  a  single  span 
into  a  group  which  can  be  handled  as  a  unit,  giving  rise  to  the 
so-called  "unit  frame".  These  units  are"  obviously  not  so 
adaptable  to  a  great  variety  of  conditions  as  single  independent 
bars,  but  .their  advantages  are  considerable  and  they  are  being 
used  to  quite  an  extent.  Fig.  94  illustrates  one  such  type  of 


e"t»o  section 


FIG.  94.— Unit  Frame. 


construction  manufactured  by  the  Unit  Concrete  Steel  Frame 
Co.  of  Philadelphia,  and  has  been  used  in  several  buildings. 
Some  of  the  transverse  slab  rods  pass  through  the  upper  ends 
of  the  stirrups  as  shown. 


258 


BUILDING  CONSTRUCTION. 


[Cn.  VII. 


Fig.  95  illustrates  a  kind  of  unit  reinforcement  on  the  Bertine 
system  and  used  in  the  warehouse  of  the  Bush  Terminal  Co., 
Brooklyn.*  Round  rods  are  used  and  tied  together  by  round 
steel  stirrups. 


FIG.  95.— Unit  Frame. 

In  all  the  examples  here  given  it  is  to  be  noted  that  the 
differences  of  detail  refer  almost  entirely  to  the  method  of 
caring  for  the  shearing  stresses  and  in  handling  the  reinforcing 
members.  The  beam  and  slab  arrangement  is  used  in  all. 

A  system  of  construction  quite  radically  different  from  the 
foregoing  is  shown  in  Fig.  96,  called  the  "  mushroom  "  system, 
devised  by  Mr.  C.  A.  P.  Turner.  No  beams  or  ribs  are  used, 
the  loads  being  transmitted  from  floor-slab  directly  to  the 
column.  The  reinforcement  is  essentially  radial  and  the  column 
is  enlarged  at  the  top  to  increase  the  circumference  at  the  line 
of  maximum  stress  in  the  slab.  The  floor  is  of  uniform  thick- 
ness throughout. 

To  a  certain  extent  this  type  of  construction  follows  the 
natural  lines  of  stress  more  closely  than  the  rectangular  ribbed- 
panel  type;  it  is  best  adapted  to  large  areas  with  few  large 
openings. 

The  analysis  of  stresses  in  this  system  may  be  made 
approximately  by  the  application  of  the  method  given  in  Art. 
150,  Chapter  VI,  and  Plates  X  and  XI.  In  applying  this 
method  to  a  continuous  floor  like  the  "  mushroom "  system, 
an  estimate  must  first  be  made  of  the  position  of  the  "  line  " 
of  inflexion  with  reference  to  the  column.  Noting  that  the 
point  of  inflexion  of  a  beam  fixed  at  the  ends  and  uniformly 

*  Eng.  Record,  Vol.  53,  1906,  p.  36. 


§  168.] 


FLOOR  AND  COLUMN  DESIGN. 


259 


loaded  is  about  one-fifth  the  span  length  from  the  end,  a  suf- 
ficiently close  estimate  of  the  "line"  of  inflexion  can  be  made. 
It  will  evidently  be  nearer  the  column  than  if  the  support  were 
a  continuous  wall.  Having  estimated  the  line  of  inflexion  the 
area  within  may  be  treated  roughly  as  a  circular  plate  loaded 
with  the  given  uniform  load  on  its  area  and  a  vertical  load 
along  its  periphery  equal  to  the  remaining  part  of  the  load 


TOP  VIEW  OF  COLUMN  FRAMING 


8,  l?g  Round  .Rods 


%  Round  Rods,  6*C.  to  C. 
PLAN  SHOWING  FLOOR  REINFORCEMENT. 


SECTIONAL  ELEVATION 

FIG.  96.— The  "Mushroom"  System. 

tributary  to  the  column.  The  diagrams  then  apply  directly. 
Thus,  suppose  the  columns  are  spaced  16  ft.  apart  and  are  20 
ins.  in  diameter  at  their  upper  ends.  Suppose  the  load  to  be 
150  lbs/in2  over  the  entire  area.  With  columns  16  ft.  apart,  the 
diagonal  spacing  will  be  about  22.5  ft.  The  line  of  inflexion 
will  probably  not  be  less  than  3  ft.  nor  more  than  4  ft.  from 
the  column  centre.  Call  it  3.5  ft.  The  area  of  this  circle  will 


260  BUILDING  CONSTRUCTION.  [Cn.  VII. 

be  38.5  sq.  ft.  The  area  of  the  entire  square  tributary  to  the 
column  is  16x16  =  256  sq.  ft.  Hence  the  total  load  applied 
along  the  periphery  will  be  (256 -  38.5)  X 150  =  32, 600  Ibs., 
which  will  be  equal  to  1480  Ibs.  per  lineal  ft.  From  Plates 
X  and  XI  the  value  of  MI  and  M2  are  found  to  be :  (a)  for 
the  direct  load  of  150  lbs/ft2,  M1  =  280  ft.-lbs.,  M2  =  1040  ft.- Ibs.; 
(6)  for  the  peripheral  load  of  1480  lbs/ft,  71^  =  2960  ft. -Ibs., 
M2=8650  ft.- Ibs.  If  ri  had  been  assumed  at  4  ft.  the  values 
of  M2  would  have  been  1500  and  9180  ft.-lbs.,  respectively. 
An  increase  in  column  diameter  to  30  ins.  would  reduce  the 
moments  M2  to  about  980  and  6600  ft.-lbs.  respectively,  as- 
suming a  value  of  ri  of  4  ft. 

In  the  illustrations  shown  the  columns  have  been  mainly 
reinforced  by  longitudinal  rods.  Various  types  of  banded  or 
hooped  columns  are  used  more  or  less,  but  usually  in  connec- 
tion with  longitudinal  reinforcement.  From  the  discussion  of 
Chapter  IV  it  would  seem  that  large  amounts  of  hooped  rein- 
forcement should  not  be  counted  upon  too  greatly  in  the  strength 
of  the  column.  In  some  forms  the  columns  are  banded  with 
spirally  wound  hooping,  as  in  the  Considere  column,  in  others 
flat  steel  is  used  in  riveted  or  welded  hoops.  Expanded  metal 
is  also  used  by  wrapping  around  longitudinal  bars. 

169.  Footings. — The  problem  of  the  design  of  footings  is 
in  general  the  same  as  that  of  floors.  On  account  of  the  heavy 
concentrated  loads  and  the  large  unit  upward  pressures  of  the 
earth  against  the  footings  the  beam  construction  will  be  rela- 
tively heavy.  The  beams  will  be  short  and  deep  and  will 
require  special  attention  to  provide  against  excessive  shearing 
and  bond  stresses.  For  single  footings  of  ordinary  size  a  single 
symmetrical  slab  is  most  convenient.  For  larger  footings  and 
for  footings  carrying  more  than  one  column,  a  combination  of 
beam  and  slab,  similar  to  floor  construction,  is  often  most 
economical. 

It  is  difficult  to  calculate  accurately  the  stresses  in  a  square 
footing,  but  assumptions  may  be  made  which  will  simplify 
the  problem  and  give  results  well  on  the  safe  side  (see  Fig.  97). 


§  169.] 


FOOTINGS. 


261 


As  a  general  principle  the  pressures  should  be  carried  as  di- 
rectly as  possible  from  the  extremities  to  the  centre.  Two 
sets  of  main  reinforcing  rods  aa'  and  66'  will  then  be  used 
as  shown  in  the  figure.  The  reinforcing  of  the  remaining  cor- 
ners can  best  be  done  by  sets  of  diagonal  rods  dd' '.  If  these 
cannot  cover  the  area,  then  a  few  short  cross-rods  may  be 
used.  Reinforced  in  this  way  the  total  pressure  on  the  area 
ABCD  may  be  assumed  to  be  carried  to  the  line  BC,  where 
the  bending  moment  and  shear  will  be  a  maximum.  Figured 
as  a  free  cantilever  the  resulting  stresses  will  be  higher  than 
actually  exist.  If  the  entire  square  be  reinforced  by  rods  in 


e  d 


*  A 


b' 
[«) 


d' 
0 

FIG.  97. 


(b) 


two  directions  only,  as  ee> ',  then  a  considerable  part  of  such 
rods  in  the  corners  of  the  square  are  ineffective. 

The  method  of  analysis  of  Art.   150,  Chapter  VI,  may  also 
be  applied  to  this  problem. 

In  the  case  of  cantilever  beams  such  as  in  footings  the  r 
maximum  shearing  stress  is  near  the  centre  where  the  maxi- 
mum moment  occurs.  Shear  cracks  tend  to  form  on  the  dotted 
curved  lines,  Fig.  97  (6).  Bent  rods,  if  used,  must  be  bent  up 
just  outside  the  column,  and  not  at  the  end  of  the  beam,  and 
stirrups  must  be  spaced  closely  at  this  point.  The  beam  being 
short  it  may  require  special  attention  to  bond  stress. 

For  large  individual  footings  a  beam  and  slab  may  be  eco- 
nomical.   To  secure  the  benefit  of  a  T  section  and  to  give 


262 


BUILDING   CONSTRUCTION. 


[Cn.  VII. 


a  flat  upper  surface  the  beam  may  be  placed  under  the  slab 
as  shown  in  Fig.  98.  This  arrangement  requires  some  atten- 
tion as  to  connection  of  slab  to  beam,  as  the  upward  pres- 
sure against  the  slab  tends  to  pull  it  away  from  the  beam.  The 
use  of  an  extra  horizontal  rod  in  the  top  of  the  main  beam 


I 


FIG.  98. 

bonded  by  stirrups  will  give  a  thoroughly  good  anchorage  for 
the  transverse  rods  of  the  slab.  For  still  larger  areas  a  sys- 
tem of  girders  and  beams  may  be  adopted  constituting  a  floor 
reversed  as  to  loads. 

170.  Walls  and  Partitions. — The  reinforcing  of  these  parts 
is  largely  for  the  purpose  of  preventing  cracks  or  of  localizing 
them  to  desired  lines.  Where  lateral  pressures  occur,  of  course 
the  beam  action  must  be  considered.  Walls  are  usually  3-6 
inches  thick  and  reinforced  both  ways  with  J-  to  J-in.  rods, 
spaced  about  2  feet  apart. 


CHAPTER  VIII. 

ARCHES. 

171.  Advantages  of  the  Reinforced  Arch.— If  the  loads  on 
an  arch  were  all  fixed  loads,  it  would  be  possible  in  any  case 
to  construct  an  arch  ring  so  that  the  resultant  pressure  at 
all  sections  would  intersect  the  centre  of  gravity  of  the  sec- 
tion. The  compressive  stress  at  any  section  would  then  be 
uniformly  distributed  over  the  section,  and  the  arch  would  be 
proportioned  only  for  this  uniform  compression.  The  "line 
of  pressure"  would  lie  at  the  axis  of  the  arch  throughout. 
If,  however,  the  arch  ring  is  not  made  to  fit  the  "  line  of  pres- 
sure", or  if '  part  of  the  load  is  a  live  load,  then  the  resultant 
pressure  will  not  in  general  coincide  with  the  axis  of  the  arch. 
There  will  exist  both  bending  and  direct  compression.  If  the 
resultant  pressure  and  its  position  are  known,  the  analysis  of 
the  stresses  at  any  section  is  made  in  accordance  with  the 
method  explained  in  Arts.  80-85,  Chapter  III. 

In  ordinary  masonry  or  concrete  arches  tensile  stresses 
are  not  permissible.  The  ring  must  therefore  be  designed  so 
that  the  line  of  pressure  will  not  pass  outside  the  middle  third. 
In  reinforced  arches  this  limitation  does  not  exist.  The  arch 
rib  is  a  beam,  and  if  properly  reinforced  it  may  carry  heavy 
bending  moments  involving  tensile  stresses  in  the  steel. 

Theoretically  the  gain  in  economy  by  the  use  of  steel  in 
a  concrete  arch  is  not  great.  If  the  pressure  line  does  not 
depart  from  the  middle  third,  the  steel  reinforces  only  in  com- 
pression and  in  this  respect  is  not  as  economical  as  concrete. 
If  the  line  of  pressure  deviates  farther  from  the  centre,  result- 
ing in  tensile  stresses  in  the  steel,  the  conditions  are  such  that 

263 


264  ARCHES.  [On.  VIII. 

those  stresses  must  be  provided  for  by  the  use  of  the  steel  at 
very  low  working  values.  That  is  to  say,  the  direct  compres- 
sion in  the  arch  is  so  large  a  factor  that  the  limiting  stresses 
in  the  concrete  will  always  result  in  very  small  unit  tensile 
stresses  in  the  steel  where  any  tension  exists  at  all. 

Practically  the  value  of  reinforcement  is  very  considerable. 
It  renders  an  arch  a  much  more  secure  and  reliable  structure, 
it  greatly  aids  in  preventing  cracks  due  to  any  slight  settlement, 
and  by  furnishing  a  form  of  construction  of  greater  reliability 
makes  possible  the  use  of  working  stresses  in  the  concrete 
considerably  higher  than  is  usual  in  plain  masonry.  Further- 
more, in  long-span  arches  where  the  dead  load  constitutes  by 
far  the  larger  part  of  the  load,  any  possible  increase  in  average 
working  stress  counts  greatly  towards  economy.  It  affects 
not  only  the  arch  but  the  abutments  and  foundations. 

172.  Methods  of  Reinforcement. — The  reinforcement  of 
arches  is  arranged  in  various  ways.  Since  the  arch  is  a  beam 
subject  to  either  positive  or  negative  bending  moments  it  is 
essential  that  it  should  be  reinforced  on  both  sides,  but  the 
shearing  stresses  due  to  beam  action  are  relatively  small,  so 
that  little  is  needed  in  the  way  of  web  reinforcement.  The 
arch  is  also  subjected  to  heavy  compression,  so  that  it  is  desir- 
able that  the  inner  and  outer  reinforcement  be  tied  together, 
somewhat  as  in  a  column,  although  in  this  case  the  necessity 
therefor  is  much  less. 

A  large  proportion  of  the  arches  which  have  been  con- 
structed have  been  built  according  to  some  one  of  the  various 
"systems"  that  have  been  devised.  The  most  important  of 
these  systems  are  the  Monier  and  the  Melan.  In  the  Monier 
system,  invented  about  1865,  the  reinforcement  consists  of 
wire  netting,  one  net  being  placed  near  the  intrados  and  one 
near  the  extrados.  The  longitudinal  wires  are  made  smaller 
than  those  following  the  arch  ring,  as  they  serve  only  to  aid 
in  equalizing  the  load  and  in  preventing  cracks.  A  large  num- 
ber of  bridges  have  been  built  in  Europe  on  this  system. 

In  the  Melan  type,  invented  about  1890,  the  steel  is  in 


§  173.]  GENERAL  METHOD  OF  PROCEDURE.  265 

the  form  of  ribs  of  rolled  I  sections,  or  of  built-up  lattice  gir- 
ders, which  are  spaced  two  to  three  feet  apart.  The  flanges 
constitute  the  principal  reinforcement,  but  the  web  enables 
the  steel  frame  to  be  self-supporting  and  to  carry  shearing 
stresses,  and  in  the  open  lattice  type  it  furnishes  a  good  bond 
with  the  concrete.  The  Melan  arch  has  been  built  extensively 
in  this  country,  largely  under  the  direction  of  Mr.  Edwin 
Thacher. 

Many  arches  are  now  being  constructed  in  which  reinforcing 
bars  of  any  satisfactory  form  are  employed  without  reference 
to  any  particular  system,  being  used  in  accordance  with  the 
requirements  of  the  case.  The  problem  of  reinforcement  is 
quite  as  simple  as  in  a  beam,  after  the  moments  and  thrusts 
in  the  arch  have  been  found. 

ANALYSIS   OF   THE    ARCH. 

173.  General  Method  of  Procedure. — The  method  of  analysis 
presented  here  is  based  on  the  elastic  theory  and  is  of  gen- 
eral application  to  arches  of  variable  moment  of  inertia  and 
loaded  in  any  manner.  It  is  mainly  an  algebraic  method, 
although  certain  simple  graphical  aids  may  be  used  advan- 
tageously. It  necessarily  assumes  that  a  preliminary  design 
has  been  made  by  the  aid  of  approximate  or  empirical  rules 
or  by  reference  to  the  proportions  of  existing  arches.  This 
arch  is  then  exactly  analyzed  and  the  results  used  in  cor- 
recting the  design;  the  corrected  design  may  then  in  turn  be 
analyzed  if  it  departs  too  greatly  from  the  one  first  assumed. 
A  discussion  of  the  various  rules  for  thickness  of  crown  and 
form  of  arch  will  not  be  entered  upon  here.  For  this  infor- 
mation the  reader  is  referred  to  the  various  treatises  on  the 
arch,  and  especially  to  those  of  Professor  Cain  and  Professor  M. 
A.  Howe.  The  work  of  Professor  Howe  on  "  Symmetrical 
Masonry  Arches "  *  contains  a  very  useful  table  of  data  of 
existing  masonry  and  reinforced-concrete  arches. 

*  New  York,  1906. 


266  ARCHES.  [Cn.  VIII. 

The  analysis  of  an  arch  consists  in  the  determination  of 
the  forces  acting  at  any  section,  usually  expressed  as  the 
thrust,  the  shear  and  the  bending  moment,  at  such  section. 
The  thrust  is  here  taken  to  be  the  component  of  the  resultant 
parallel  to  the  arch  axis  at  the  given  point,  and  the  shear  is 
the  component  at  right  angles  to  such  axis.  The  thrust  causes 
simple  compressive  stresses;  the  shear  causes  stresses  similar 
to  those  produced  by  the  vertical  shear  in  a  simple  beam. 

The  method  of  procedure  will  be  to  determine,  first,  the 
thrust,  shear,  and  bending  moment  at  the  crown.  These  being 
known,  the  values  of  similar  quantities  for  any  other  section 
can  readily  be  determined.  A  length  of  arch  of  one  unit  will 
be  considered. 

174.  Thrust,  Shear,  and  Moment  at  the  Crown  (H0,V0,M0). 

Notation.  (See  Fig.  100.) 

Let  HQ  —  thrust  at  the  crown  ;^ 
VQ  =  shear  at  the  crown; 

MO  — bending  moment  at  the  crown,  assumed  as 
positive   when    causing    compression    in 
the  upper  fibres; 
N,  V,  and  M  =  thrust,  shear',  and   moment  at  any  other 

section; 
R  =  resultant  pressure  at  any  section  =  resultant 

of  N  and  7; 

ds  =  length  of  a  division  of  the  arch  ring  meas- 
ured along  the  arch  axis; 
n  =  number  of  divisions  in  one-half  of  the  arch; 
7= moment  of  inertia  of  any  section  =  /concrete 

+  n!M  (seep.  92); 
P=any  load  on  the  arch; 

x,  y— co-ordinates  of  any  point  on  the  arch  axis 
referred  to  the  crown  as  origin,  and  all 
to  be  considered  as  positive  in  sign; 
m  =  bending  moment  at  any  point  in  the  canti- 
lever, Fig.  100,  due  to  external  loads. 


§  174.]     THRUST,  SHEAR,  AND  MOMENT  AT  THE  CROWN.      267 

LekAB,  Fig.  99,  be  a  symmetrical  arch  loaded  in  any  man- 
ner with  loads  PI,  P2,  etc.  Divide  the  arch  into  an  even  num- 
ber of  divisions  (ten  to  twenty  usually),  making  the  divisions 
of  such  a  length  that  the  ratio  ds:I  will  be  constant.  This 
may  be  done  by  trial  or  by  the  more  direct  method  explained 
in  Art.  178.  Mark  the  centre  point  of  each  division  and  num- 


FIG.  99. 


ber  the  points  as  shown.  Consider  the  arch  to  be  cut  at  the 
crown  and  each  half  to  act  as  a  cantilever  subjected  to  exactly 
the  same  forces  as  exist  in  the  arch  itself,  that  is,  the  given 
loads,  the  reactions,  and  the  stresses  at  the  crown,  represented 


FIG.  100. 


by  H0)  Fo,  and  M0  (Fig.  100).    H0  is  applied  at  the  gravity 
axis. 

Let  m  =  bending  moment  at  any  point,  1,  2,  3,  etc.,  due  to 
the  given  external  loads  P  (considering  the  arch  as  two  canti- 
levers). Denote  by  mR  the  moments  in  the  right  half  and 
by  7r,L  those  in  the  left  half  of  the  arch.  All  the  moments  m 


268  ARCHES.  [Cn.  VIII. 

will  be  negative.    The  values  of  HQ,  V0,  and  MQ  will  then 
be  given  by  the  following  equations: 

_  nlmy-  Imly 
2> 


I(mR-mL)x 
Vo~ 


In  these  equations  the  summations  ly,  ly2,  and  Ix2  are 
for  one-half  of  the  arch  only;  the  summation  Im  is  for  the 
entire  arch  and  is  equal  to  Im-^-\-  Im^  the  summation 
2(mR—niL)x  is  a  summation  of  the  products  (mR—m^x,  in 
which  mR  and  mL  are  the  bending  moments  at  corresponding 
points  in  the  right  and  left  halves  which  have  equal  abscissas  x\ 
and  the  summation  2  my  is  for  the  entire  arch,  but  since  sym- 
metrical points  have  equal  y's  this  quantity  may  be  calculated 
as  2(m^+mj^y.  A  positive  result  for  VQ  indicates  action  as 
shown  in  Fig.  100.  All  quantities  are  readily  calculated 
Distances  should  be  scaled  and  quantities  tabulated.* 

*  Demonstration.  —  Consider  the  left-hand  cantilever  of  Fig.  100.  Under 
the  forces  acting  the  point  C  will  deflect  and  the  tangent  to  the  axis  at  this 
point  will  change  direction  (the  abutment  at  A  being  fixed).  Let  Ay,  Ax, 
and  J<£  be,  respectively,  the  vertical  and  horizontal  components  of  this 
motion  and  the  change  in  angle  of  the  tangent.  Then  according  to  the  prin- 
ciples relating  to  curved  beams  1  we  have  the  values 


,      Ax=2Mx,    and      4£-JJf~    ...     (a) 

in  which  the  various  quantities  have  the  same  significance  as  in  Art.  174. 

In  like  manner,  referring  to  the  right  cantilever,  let  Ay',  Ax',  and  A$' 
represent  the  components  of  the  movement  of  C  and  the  change  of  angle  of 
the  tangent.  These  may  be  expressed  in  terms  similar  to  Eq.  (a). 

Now  evidently 

Ay=Ay',     Ax=-Axf,     and     A^=-A^f.     ....     (6) 

Furthermore,  since  ds/I  is  constant  and  likewise  E,  the  quantity  ds/EI  may 
be  placed  outside  the  summation  si?-n. 

!See  Church's  Mechanics,  or  Johnson's  Framed  Structures,  p.  236. 


§  176.]  PARTIAL  GRAPHICAL  CALCULATION.  269 

175.  Thrust,  Shear,  and  Moment  at  any  Section. — The 

values  of  HQ,  VQ,  and  MQ  having  been  found,  the  total  bend- 
ing moment  at  any  section,  1,  2,  etc.,  is 

M=m+Mo+H0y±Vox.       .    ..    .    .    (4) 

The  plus  sign  is  to  be  used  for  the  left  half  and  the  minus  sign 
for  the  right  half  of  the  arch. 

The  resultant  pressure,  R,  at  any  section  of  the  arch  is 
equal  in  magnitude  to  the  combined  resultant  of  the  external 
loads  between  the  crown  and  the  section  in  question,  and  the 
forces  HQ  and  VQ.  These  resultants  can  best  be  found  graphic- 
ally. The  thrust,  N,  is  the  component  of  the  resultant,  R, 
parallel  to  the  arch  axis  and  the  shear,  V,  is  the  component 
perpendicular  to  this  axis. 

176.  Partial  Graphical  Calculation.— Where  the  loads  are 
vertical  the  calculation  of  the  quantities  m  can  be  advan- 
tageously performed  by  means  of  .an  equilibrium  polygon  as 
follows : 

Let  AB,  Fig.  101,  represent  the  arch  axis.  The  load  line  is 
a-c-b.  Select  any  convenient  pole  0  on  a  horizontal  line  through 

Using  the  subscript  L  to  denote  left  side  and  R  to  denote  right  side  we 
then  derive  the  relations 


IMLy=-IMRy, 
IML  =-2M 

The  moment  M  may  in  general  be  expressed  in  terms  of  known  and  un- 
known quantities  thus: 

ML=mL+Ms+HQy+V<F  for  the  left  side 
and 

MR  =  mR+  M0+  H0y  —  VoX  for  the  right  side. 

Hence,  substituting  in  (c)  and  combining  terms,  and   noting  that  2M9  for 
one  half  is  equal  to  nM0i  we  have 

Smifl-  2mRx+2V0Sx  =  0,     .     .     .     .     .     .'.'".    (rf) 

ImLy+  ZmRy+  2M0Iy+  2H02y*  =  0,  ?    :.     .     .     .      (e) 


From  Eq.  (d)  is  obtained  Eq.  (2),  p.  268;   and  from  Eqs.  (e)  and  (/)  are 
obtained  Eqs.  (1)  and  (3),  noting  that  2w£,+  2mR=  Im,  and  Imi,y+  2mRij= 


270 


ARCHES. 


[Cn.  VTII. 


the  point  c,  at  the  junction  of  loads  P3  and  P4,  the  loads  adja- 
cent to  the  crown  C.  Construct  the  equilibrium  polygon  efgh, 
producing  to  i  and  k  the  segment  fg  between  loads  P3  and  P4. 
Drop  verticals  from  the  points  1,  2,  3,  etc.  The  desired  bend- 
ing moments  m,  at  the  several  points,  will  then  be  equal  to 
the  corresponding  intercepts  z2,  z3,  etc.,  on  these  verticals 
between  the  equilibrium  polygon  and  the  line  ikt  multiplied 
by  the  pole  distance  H'}  or  in  general  m=Hz. 


FIG.  101. 

Finally,  after  the  values  of  H0,  V0,  and  M0  are  found  by 
Eqs.   (1),  (2),  and  (3),  the  true  equilibrium  polygon  can  be     / 
drawn,  if  desired,  and  values  of  thrust,  shear,  and  moment  / 
at  various  points  determined  graphically.    The  true  pole  is 
found  by  laying  off  VQ  and  HQ  from  the  point  c.     The  dis- 
tance of  the  equilibrium  polygon  above  or  below  the  axis  at 
the  crown  i-s  equal  to  Mo/H0.      It  lies  above  the   axis  if  the 
result  is  positive  and   below  if  negative.      The   equilibrium 
polygon  is  then  drawn  from  the  crown  each  way  towards  the 
ends. 


§  178.]  DIVISION  OF  ARCH  RING.  271 

Where  the  loads  are  inclined  at  various  angles  it  is  still 
possible  to  use  a  graphical  process  for  getting  values  of  m, 
but  there  is  little  to  be  gained  in  such  a  case.  After  the  values 
of  HQ,  V0,  and  M 0  are  found,  however,  it  will  be  expedient  to 
draw  a  final  equilibrium  polygon,  or  line  of  pressure,  as  ex- 
plained above. 

177.  General  Observations. — The  method  of  analysis  just 
described  is  brief,  general,  and  easily  followed.     The  arithmetical 
calculations  are  not  longer  than  those  required  in  the  usual 
graphical  process,  while  the  graphical  aids  here  suggested  are 
of  the  simplest  character. 

The  loads  and  their  points  of  application  have  been  con- 
sidered apart  from  the  divisions  of  the  arch  ring,  as  the  two 
things  are  in  no  wise  related.  Where  no  spandrel  arches  are 
used  and  the  entire  load  is  applied  continuously  along  the 
arch  ring,  the  load  may  for  convenience  be  divided  to  corre- 
spond with  the  arch  divisions  and  applied  at  the  center  points, 
1,  2,  3,  etc.  This  division  is,  however,  of  no  importance,  the 
only  requirement  being  a  sufficiently  small  subdivision  of  the 
arch  ring  and  of  the  load  so  that  the  errors  of  approximation 
will  be  negligible.  Where  spandrel  arches  are  used,  the  live 
load  and  a  large  part  of  the  dead  load  will  be  applied  at  the 
centers  of  the  arch  piers.  The  weight  of  the  main  arch  ring 
may  also  be  considered  as  concentrated  at  these  same  points. 

If  calculations  are  to  be  made  for  more  than  one  loading 
it  will  be  noted  that  the  denominators  of  the  values  for  H0, 
VQ,  and  MQ  do  not  change.  The  quantities  involving  m  are 
the  only  ones  requiring  recalculation,  and  if  the  load  on  but 
one-half  of  the  arch  is  changed,  then  the  values  of  m  for  that 
half  only  need  be  recalculated.  In  the  case  of  a  symmetrical 
loading,  or  a  load  on  one-half  only,  the  calculation  of  m  is 
also  necessary  for  one-half  the  arch  only.  For  symmetrical 
loads,  7o=0. 

178.  Division  of  Arch  Ring  to  give  Constant  <5s/I. — In  most 
cases  the  depth  of  the  arch  ring  increases  from  crown  towards 
springing  line,  giving  a  variable  moment  of  inertia.    Consider- 


272  ARCHES.  [Cn.  VIII. 

ing  the  concrete  only,  the  moment  of  inertia  will  increase  as  d3 
so  that  a  comparatively  small  change  in  depth  will  cause  a 
large  change  in  moment  of  inertia.  To  maintain  ds/I  con- 
stant, the  value  of  ds  will  therefore  be  much  greater  near  the 
springing  line  than  at  the  crown,  and  hence  to  secure  the  de- 
sired accuracy  the  length  of  division  at  the  crown  will  need 
to  be  made  fairly  short.  The  value  of  ds/I  to  adopt  so  that 
there  will  be  no  fractional  division  may  be  determined  as  fol- 
lows: 


Let   ^=-T; 

i0=mean  value  of  i\ 
s  =  half    length   of   the  arch  ring  measured  along  the 

axis; 
n  =  desired  number  of  divisions  in  one-half  the  arch. 

Calculate  first  the  mean  value  of  i  for  the  half  arch  ring  by 
determining  several  values  at  equal  intervals  along  the  arch. 
Then  the  desired  value  of  ds/I  is 


(5) 
(5j 


The  value  of  ds/I  being  known,  the  proper  length  of  ds  for 
any  part  of  the  arch  ring  can  readily  be  determined.  Begin- 
ning at  the  crown,  the  length  of  the  first  division  is  deter- 
mined, then  the  second,  third,  etc.,  to  'the  end.  The  length 
of  a  division  not  being  exactly  known  beforehand,  the  value 
of  /  for  that  division  will  not  be  exactly  known,  but  the  neces- 
sary adjustment  is  very  simple. 

In  determining  the  value  of  I  the  steel  reinforcement  must 
be  duly  considered. 

179.  Temperature  Stresses.  —  For  a  rise  of  temperature  of  t 
degrees  the  increase  in  span-length,  if  the  arch  be  not  re- 
strained, would  be  equal  to  ctl,  where  c  =  coefficient  of  expan- 


§  180.]         STRESSES  DUE  TO  SHORTENING  OF  ARCH.  273 

sion  and  J  =  span.     The  restraint  of  the  abutments  induces  a 
thrust  HQ  at  the  crown  given  by  the  equation 

=  m  ctln          * 

°~  ds'2[nlyz-(ly)2\  ...... 

The  summations  refer  to  one-half  the  arch. 
Having  determined  HQ,  then 


The  bending  moment  at  any  point  is 

M  =  M0  +  H0y  ......     .     (8) 

The  thrust  and  shear  at  any  point  in  the  arch  are  found 
by  resolving  H0  parallel  and  normal  to  the  arch  axis  at  that 
point.  Graphically,  the  true  equilibrium  polygon  is  a  hori- 
zontal line  drawn  a  distance  below  the  crown  equal  to  MQ/HQ 


1  80.  Stresses  Due  to  Shortening  of  Arch  from  Thrust.  —  A 

thrust  throughout  the  arch  producing  an  average  stress  on 
the  concrete  equal  to  fc  lbs/in2  would  shorten  the  arch  span 
an  amount  equal  to  fcl/E  if  unrestrained.  This  action  develops 
horizontal  reactions  in  the  same  manner  as  a  lowering  of  tem- 


*  Demonstration. — For  temperature  stresses  J^  of  Eq.    (a),  p.  268,  is 
o  and  Ax  is 
therefore  have 


zero  and  Ax  is  equal  to  the  change  in  length  of  the  half-span,  =C—.     We 


„,,      ds     ctl 
IMLyjj=—t 

and 

2ML=Q. 

In  this  case,  there  being  no  external  loads,  m  =  Q,  and  from  symmetry, 
F0  =  0,  hence  M  =  M0+HQy.  Substituting  this  value  of  M  in  the  above 
equations  we  have 

M      V       ,      V      V     2        Ctl      El 

M02y+H0Zy2  =  —  .—, 
and 


From  these  are  readily  derived  Eqs.  (6)  and  (7). 


274  ARCHES.  [On.  VIII. 

perature.  The  value  of  the  resulting  reactions,  or  the  crown 
thrust,  may  then  be  found  by  substituting  fcl/E  for  ctl  of 
Eq.  (6).  There  results 

I_  fcln 

'22' 


The  moments  at  crown  and  elsewhere  are  given  by  Eqs.  (7) 
and  (8),  using  the  value  of  HQ  from  Eq.  (9). 

The  thrusts  and  moments  due  to  arch  shortening  will  not 
usually  be  large.  They  may  be  applied  as  corrections  to  the 
thrusts  and  moments  found  before. 

181.  Deflection  of  the  Crown.  —  The  downward  deflection 
of  the  crown  under  a  load  is  given  by  Eq.  (a),  p.  268.  It  is 


Jj/=  -    jZMx  .......     (10) 

If  M  is  not  determined  for  all  points,  use  the  value  of  M 
from  Eq.  (4),  deriving 


.     (11) 


The  summations  are  for  one-half  only. 

The  rise  of  crown  due  to  an  increase  of  temperature  is  ob- 
tained from  Eq.  (11)  by  substituting  from  Eqs.  (6)  and  (7). 
There  results 

ctl  nZxy-  Ixly 


182.  Unsymmetrical  Arches.  —  If  the  arch  is  unsymmet- 
rical  the  value  of  ds/I  should  be  made  constant  for  the  entire 
arch,  and  the  number  of  divisions  made  even  as  before.  The 
central  point  of  the  arch  with  reference  to  the  number  of  divi- 
sions may  then  be  taken  as  the  crown,  and  the  X-axis  made 
tangent  to  the  arch  at  this  point.  The  two  halves  of  the  arch 
are  now  unlike  and  all  terms  resulting  from  substitution  in 
Eq.  (c),  p.  269,  must  be  retained.  Explicit  formulas  for  HQ, 


§  183.1         APPLICATION  OF  THE  PRECEDING  THEORY. 


275 


V0,  and  MQ  are  very  cumbersome,  but  the  three  equations 

derived  from  (c)  are  as  follows  : 

(ILx- IRx)M0  +  (2Lxy-  2Rxy)H0 -f  2&VQ  =  ZRmx-  2Lmx,  (13) 

.  .  (14) 
.  .  (15) 

Where  no  subscript  is  used  the  summation  is  for  the  entire 
arch.  Numerical  values  of  the  coefficients  of  M 0,  H0,  and  VQ 
should  be  obtained  and  the  equations  then  solved. 

183.  Application  of  the  Preceding  Theory. — Example  1. — 
An  arch  ring  will  be  assumed  of  the  dimensions  shown  in 
Fig.  102.  Span  length  with  reference  to  the  axis  =30  ft., 
rise  =  8  ft.  Thickness  at  crown  =  1  ft.,  at  springing  lines  = 
1  ft.  6  in.  For  a  small  arch  such  as  this  great  accuracy  is  not 
needed,  hence  a  small  number  of  divisions  may  be  used.  The 
number  will  be  four  for  each  half.  These  divisions  are  deter- 
mined so  that  ds/I  is  constant.  The  loads  are  applied  at 
the  centre-points  1,  2,  3,  4,  and  are  assumed  to  be  somewhat 
inclined  (excepting  loads  P4  and  P5),  the  several  vertical  and 
horizontal  components  being  as  given  in  the  figure. 

TABLE  A. 
CALCULATIONS  FOR  #0,  70,   AND  M0. 


1 

2 

3 

4 

5 

.6 

7 

8 

9 

Point. 

X 

y 

a? 

2/2 

rriL 

mR 

(mi+mtfty 

(mR-mL)x 

1 
2 
3 
4 

1.55 
4.90 
8.45 
12.85 

.09 
.68 
2.10 
5.35 

2.40 
24.01 
71.40 
165.12 

.01 
.46 
4.41 

28.62 

0 

-  13,840 
-  46,800 
-116,680 

0 
-     8,640 
-  29,560 
-  75,490 

0 
-  15,300 
-  160,400 
-1,028,100 

0 
+  25,500 
+  145,700 
+  529,300 

I 

S.22 

262.93 

33.50 

-1,203,800 

+  700,500 

-291,010 

Spring- 
ing. 

15.00 

8.00 

-180,820 

-120,640 

F      m     '         77      4(-1203800)- (-291010X8.22)         18Oqn]K 
Eq.  (1)  gives  H0=  2[(8.22)2 -4X33.50]  =  +  18'23°  lbs' 

Ibs. 


Eq.  (2)  gives  ^= 

-291010+2X18230X8.22 
Eq.  (3)  gives  M0= =  -1,090  ft-lbs. 


276 


ARCHES. 


[Cn.  VIII. 


TABLE  B. 
BENDING  MOMENTS,  THRUSTS,  AND  ECCENTRIC  DISTANCES. 


1 

2 

3 

4 

5 

6 

7 

$ 

9 

Bending  Moment  M. 

Thrusts. 

Eccentric 
Distances. 

Point. 

H0y 

Vox 

Left. 

Right. 

Left. 

Right. 

Left. 

Right. 

1 

1,640 

2,060 

+  2,610 

-1,520 

18,450 

18,640 

+  .14 

-.08 

2 

12,400 

6,530 

+  4,000 

-3,860 

19,580 

19,310 

+  .21 

-.20 

3 

38,300 

11,260 

+  1,650 

-3,620 

22,050 

20,770 

+  .07 

-.17 

4 

97,500 

17,120 

-3,100 

+  3,850 

28,800 

24,970 

-.11 

+  .15 

Spring- 
ing. 

145,900 

20,000 

-16,070 

+  4,140 

28,800 

24,970 

-.56 

+  .17 

The  calculations  of  the  several  quantities  in  the  formulas 
for  //0,  To,  and  M0  (p.  268)  are  given  in  Table  A.  The  co- 
ordinates x,  y  of  the  several  points  are  given  in  cols.  2  and 
3;  then  x2  and  y'2  in  cols.  4  and  5;  then  in  cols.  6  and  7  are 
given  the  quantities  mL  and  mR,  considering  each  half-arch  a 
cantilever.  These  are  readily  calculated.  Thus,  on  the  left, 
for  point  1,  w  =  0;  for  point  2,  w  = 4130 X  (4.90 -1.55)  =  13,840; 
for  point  3,  m  =  4130  X  (8.45  - 1 .55)  +  5035  X  (8.45  - 4.90)  +  310  X 
(2.10 -.68) -46, 800;  and  for  point  4,  m  =  4130  X  (12.85 -1.55)  + 
5035 X  (12.85-4.90)  +  5950  X  (12.85  -8.45)  +  310 X  (5.35  -  .68)  + 
725(5.35 -2.10)  =  11 6,680.  The  value  of  m  at  the  springing 
line  is  also  calculated  and  placed  in  this  table  for  future  use. 
The  moments  on  the  right  are  similarly  found.  All  moments  m 
are  negative.  In  cols.  8  and  9  are  then  given  the  products 
(mr,+mR)y  and  (mR-mL)x. 

Substituting  in  Eqs.  (1),  (2),  and  (3),  p.  268,  there  are  ob- 
tained the  values  for  HQ,  T"o,  and  A/o  given  below  the  table. 

The  values  of  the  bending  moments,  thrusts,  and  shears  at 
any  point  may  now  be  found  either  graphically  or  algebraically. 
The  force-diagram  method  will  be  much  the  better  for  obtaining 
thrusts  and  shears;  the  moments  may  then  be  obtained  either 


§  183.]        APPLICATION  OF  THE   PRECEDING   THEORY. 


27T 


by  constructing  an  equilibrium  polygon  or  by  the  application 
of  Eq.  (4),  p.  269. 

In  Fig.  102  the  graphical  construction  is  given.  The  load- 
line  is  a-c-b.  The  true  pole  is  found  by  laying  off  F0  =  + 1330 
from  point  c  (at  the  junction  of  the  loads  adjacent  to  the 
crown,  P4andP5);  then  H0  =  18,230  horizontally  to  0.  The 


force  diagram  is  drawn  and  then  the  equilibrium  polygon,, 
beginning  at  the  crown  and  drawing  the  segment  1  —  1  at  a 
distance  below  the  crown  equal  to  1090/18230  =  .06  ft.  The 
resultant,  R,  acting  at  any  section  may  be  scaled  from  the 
force  polygon,  and  the  moment  at  any  point  will  be  equal  to 
this  resultant  multiplied  by  the  perpendicular  distance  at  that 
point  from  the  arch  axis  to  the  equilibrium  polygon.  For  ex- 
ample, the  bending  moment  at  point  g  is  equal  to  the  force  Od 
multiplied  by  the  arm  gk.  The  tangential  component  of  the 


278  ARCHES.  [Cn.  VIII. 

resultant  R  (the  true  thrust  Ar)  may  be  found  by  resolving  the 
force  R  parallel  and  normal  to  the  arch  axis  at  the  point  in 
question.  In  most  cases  the  thrust  N  may  be  taken  as  equal 
to  R.  The  shear  V  will  be  the  normal  component  of  R'}  it  will 
not  usually  require  consideration. 

Table  B  contains  calculated  values  of  moments  and  eccen- 
tric distances  for  points  1,  2,  3,  4  and  the  springing  lines.  The 
moments  are  calculated  from  the  formula  (Eq.  (4))  M  = 
m  +  Mo  +  Hoy±V0x.  The  quantities  m  are  obtained  from 
Table  A,  cols.  6  and  7.  The  thrusts  are  scaled  from  the  force 
polygon,  being  in  each  case  the  thrust  on  the  abutment  side  of 
the  point  in  question.  The  eccentric  distances  are  equal  to 
the  moments  divided  by  the  thrusts;  they  are  of  use  in  cal- 
culating stresses  in  the  arch.  Obviously  the  bending  moment 
at  any  other  point,  such  as  g,  may  be  calculated  in  the  same 
way  as  those  here  given. 

184.  Example  2.  (Fig.  103.) — For  another  example  an  arch 
will  be  assumed  of  100-ft.  span  and  20-ft.  rise;  thickness  at 
crown  =  30  in.,  thickness  at  springing  line  =  42  in.  It  will  be 
assumed  that  the  roadway  is  supported  on  spandrel  piers  10  ft. 
apart,  thus  concentrating  most  of  the  load  at  points  10  ft. 
apart  as  shown;  the  weight  of  the  arch  ring  will  also  be  as- 
sumed as  applied  at  these  points.  The  loads  as  given  represent 
an  arch  with  live  load  on  the  left  half  only.  The  half  arch  is 
divided  into  ten  divisions,  making  ds/I  constant.  The  loads  in 
this  case  are  vertical,  so  that  the  graphical  method  may  be  used 
to  advantage  in  determining  the  cantilever  moments  m.  The 
load  line  a-c-b  is  drawn  and  a  pole  0'  selected  on  a  horizontal 
line  through  c  at  the  center  of  the  crown  load  PS.  The  pole 
distance  is  H.  An  equilibrium  polygon,  efgh,  is  then  drawn, 
and  the  moments  m  will  be  equal  to  the  intercepts,  z,  from  this 
polygon  to  the  horizontal  line  ik,  multiplied  by  H.  These 
moments,  and  the  remainder  of  the  calculations  for  H0,  F0,  and 
MQ  are  given  in  Table  C.  The  true  pole  0  is  then  found  as 
before  and  the  correct  equilibrium  polygon  drawn.  The 
thrusts  are  then  scaled  from  the  force  polygon,  and  the  eccentric 


§  184.]         APPLICATION  OF  THE  PRECEDING  THEORY.  279 


280 


ARCHES. 


[CH.  VIII. 


distances  from  the  equilibrium  polygon.  These  are  given  hi 
Table  D,  together  with  resulting  bending  moments.  The  bend- 
ing moments  may  also  be  calculated  as  done  in  Example  1. 

TABLE  C. 
CALCULATIONS  FOR  HQ,  F0,  AND  M0. 


1 

2 

3 

4 

5 

6 

7 

8 

9 

Point 

X 

y 

*2 

y2 

mL 

mR 

(mL  +  mR)y 

(mR-mL)x 

1 

2.07 

.07 

4.3 

.00 

-       13,500 

-       13,500 

2,000 

2 

5.96 

.30 

35.5 

.09 

—      38,700 

-       38,700 

23,000 



3 

10.00 

.72 

100.0 

.52 

-       65,000 

-       65,000 

94,000 

4 

L4.18 

1.40 

201.1 

1.96 

-     148,600 

-     127,700 

-       387,000 

+       296,000 

5 

18.43 

2.37 

339.7 

5.62 

-     233,600 

-     191,400 

-    1,007,000 

+      777,000 

6 

23.06 

3.77 

531.8 

14.21 

-     369,000 

-     288,400 

-    2,479,000 

+    1,859000 

7 

28.06 

5.64 

787.4 

31.81 

-     539,000 

-     408,400 

-    5,348,000 

+   3,665,000 

8 

33.60 

8.23 

1129.0 

67.73 

-     781,400 

-     577,400 

-11,183,000 

+    6,854,000 

9 

39  .  60 

11.80 

1568.2 

1  39  .  24 

-1,075,400 

-     781,400 

^21,910,000 

+  11,642.000 

10 

46.  40 

16.&0 

2153.0 

282.24 

-1,511,000 

-1,083,000 

-43,580,000 

+  19,859,000 

2 

51  .  10 

68.50.0 

543  .  42 

-86,013,000 

+  44,952,000 

-8,350,100 

IPX  (-86013000)-  (-8350100)X51.10 

2.[  (51.10)'-  10X543.42] 
44952000 
2X6850  ~  -«V8(Jlbs. 

-8350100  +  2X76760X61.10.  + 


° 


TABLE  D. 
THRUSTS,  ECCENTRIC  DISTANCES,  AND  MOMENTS. 


1 

2 

3 

4 

6 

6 

7 

Thrusts. 

Eccentric  Distances. 

Bending  Moments. 

Point. 

Right. 

Left. 

Left. 

Right. 

Left. 

Right. 

1 

76,800 

77,400 

+  .31 

+  .13 

+  23,700 

+    10,100 

2 

76,800 

77,400 

+  .38 

-.13 

+  29,200 

-  10,100 

3 

78,600 

78,800 

+  .61 

-.22 

+  48,000 

-  17,300 

4 

78,600 

78,800 

+  .39 

-.53 

+  30,700 

-  41,700 

5 

78,600 

78,800 

+  .44 

-.57 

+  34,700 

-  45,000 

6 

82,700 

81,500 

+  .26 

-.61 

+  21,200 

-  49,700 

7 

82,700 

81,500 

+  .14 

-.52 

+  11,400 

-  42,400 

8 

89,300 

85,300 

-.15 

-.36 

-13,400 

-  30,700 

9 

89,300 

85,300 

-.16 

+  .23 

-14,300 

+    19,600 

10 

98,600 

90,700 

-.44 

-1-  .88 

-43,400 

+   80,000 

Springing 

98,600 

90,700 

-.21 

+  1.67 

-20,700 

+  152,000 

§  185  ]  MAXIMUM  STRESSES.  281 

Temperature  Stresses. — Suppose  in  Ex.  2  it  is  desired  to 
know  the  thrust  and  bending  moment  at  the  crown  due  to  a 
rise  of  temperature  of  30°.  Eqs.  (6)  and  (7),  Art.  179,  will  be 
used.  Assume  E =2,000,000  Ibs /in2  =  288,000,000  Ibs/f t2.  Sup- 
pose the  value  ds/I,  in  foot-units,  is  3.1.  Then  from  Eq.  (6) 

288,000,000     .000006X30X100X10 

3.1  '    2(10  X  543  -(51.1)2)    ~2970    °S'' 

Mo  =  -  2970  X       -  =  - 15,200  ft-lbs. 


The  equilibrium  polygon  is  a  horizontal  line  drawn  a  distance 
below  the  crown  equal  to  15200/2970=5.11  ft.  The  moment 
at  any  point  is  equal  to  the  thrust  HQ  multiplied  by  the  vertical 
distance  from  such  point  to  this  equilibrium  polygon.  At  the 
springing  line,  M=HQX  (20 -5.11)  =2970X14.89=44,200  ft- 
lbs.  This  may  also  be  calculated  by  Eq.  (8). 

Stresses  Due  to  Shortening  of  Arch. — The  modification  of 
the  thrust  due  to  the  compressive  deformations  of  the  arch 
ring  is  found  by  Eq.  (9).  The  average  compressive  stress  at 
any  section  is  found  by  dividing  the  thrust  at  that  section  by 
the  area  of  the  transformed  section  of  arch  ring.  This  is  nearly 
uniform  throughout  the  arch  and  equal  to  about  150  lbs/in2. 
Then, 

J_     150X144X100X10 
3.1X2[10X543-(51.1)2] 

This  thrust  is  equal  to  42%  of  the  thrust  due  to  temperature 
change,  already  found.  The  resulting  moments  and  stresses 
will  then  be  42%  of  those  due  to  temperature  change.  They 
will  be  of  opposite  sign. 

185.  Maximum  Stresses  in  the  Arch  Ring.— From  the 
values  of  thrust,  moment,  and  eccentric  distance,  as  given  in 
Tables  B  and  D,  the  stresses  in  the  concrete  and  steel  can  be 
found  at  any  section  of  the  arch,  as  explained  in  Chap.  Ill  and 
also  in  Art.  147,  Chap.  VI.  The  maximum  value  of  fibre  stress 


282  ARCHES  [Cn  VIII. 

will  be  where  the  sum  of  the  stresses  due  to  thrust,  N,  and 
moment,  M,  is  a  maximum.  This  will  not  in  general  be  where 
either  the  thrust  or  the  moment  is  a  maximum;  but  as  the 
thrust  varies  slowly  along  the  arch  ring  the  maximum  stress  will 
occur  very  near  to  the  .point  of  maximum  moment. 

The  position  of  live  load  causing  maximum  moment  at  any 
point  will  differ  in  arches  of  different  proportions.  In  designing 
an  arch  it  is  sufficient  generally  to  determine  the  maximum 
stresses  at  the  crown,  the  haunch,  and  the  springing  line.  This 
will  require  several  different  positions  of  the  live  load.  For  the 
crown  the  maximum  positive  moments  are  caused  when  a 
short  length  of  the  arch  (one-fourth  to  one-third)  at  the  center 
is  loaded,  and  the  maximum  negative  moments  when  the  re- 
maining portions  are  loaded.  The  maximum  positive  and 
negative  moments  at  the  haunch  (about  the  J  point)  are  caused 
when  the  arch  is  loaded  about  two-thirds  the  span  length  and 
one-third  the  span  length  respectively.  The  same  loading  will 
give  practically  the  maximum  moments  at  the  springing  lines. 

These  conditions  make  it  desirable  to  analyze  the  arch  for 
various  assumed  loadings  about  as  follows:  full  load;  one-third 
of  span  loaded;  two-thirds  of  span  loaded;  center  third  loaded; 
and  end  thirds  loaded.  In  the  case  of  large  and  important 
structures  it  may  be  found  desirable  to  place  the  loads  some- 
what differently  than  here  indicated.  A  complete  arid  exact 
solution  can  readily  be  made  by  analyzing  the  arch  for  a  load 
of  unity  at  each  load-point  of  one-half  of  the  arch.  Influence 
lines  can  then  be  drawn  for  moment  or  fibre  stress  and  the 
exact  maximum  values  determined. 

1 86.  Illustrative  Examples  of  Arch  Design. — Fig.  104 
shows  a  longitudinal  section  and  part  plan  of  a  bridge  at  Grand 
Rapids,  Mich.*  It  consists  of  five  spans  of  lengths  from  79  to 
87  feet.  The  reinforcement  is  composed  of  IJ-in.  Thacher  bars 
spaced  14  in.  apart  near  both  the  extrados  and  in  trad  os.  Each 
pair  is  connected  by  f-in.  connecting-rods  spaced  4  ft.  apart. 

*Eng.  News,  Vol.  LIT,  1904,  p.  490. 


§  186.] 


EXAMPLES  OF  ARCH  DESIGN. 


283 


Cast  Iron  Elp» 


LONGITUDINAL  SECTION 

FIG.  104. — Arch  Bridge  at  Grand  Rapids,  Mich. 


H  Ba'S.S'ctrs 


HALF  ELEVATION 


FIG.  105. — Arch  Bridge  on  the  Chicago  &  Eastern  Illinois  R.R. 


284 


ARCHES. 


[Cn.  VIH. 


Fig.  105  illustrates  a  38-ft.  span  reinforced  arch  on  the 
C.  &  E.  I.  R.R.*    Johnson  corrugated  bars  were  used. 


FIG.  106. 

Figs.  106  and  107  show  the  details  of  a  design  for  a    concrete 
viaduct  at  Milwaukee,  Wis.f    The  design  was  made  by  the 

*  R.R.  Gaz.,  April  13,  1906,  p.  390. 
t  Eng.  News,  Vol.  LVH,  1907,  p.  178. 


§  186.] 


EXAMPLES  OF  ARCH  DESIGN. 


285 


Concrete-Steel  Engineering  Co.,  of  New  York  City,  and  is  a 
typical  Melan  arch.  The  reinforcement  consists  of  built-up 
ribs  of  3"X3"Xf"  angles  connected  by  lattice  bars  2"Xi". 


18.1  >4"  Diam.Thacher  Bare.  12  C.to  C. 


98.1)4  "  Thacher  Bars,  8  C.to  C. 


HALF  VERTICAL  SECTION 

AT  CENTER  LINE 

OF  PIER 


-12" 


HALF  VERTICAL  SECTION  A-B 

FIG.  107. 


The  ribs  are  spaced  3  ft.  apart.  These  ribs  are  designed  to 
carry  the  entire  bending  moment  at  a  stress  of  18,000  lbs/in2. 
The  stress  in  the  concrete  was  limited  to  500  lbs/in2  in  com- 


286 


ARCHES. 


§  186.] 


EXAMPLES  OF  ARCH  DESIGN. 


287 


pression,  or  600  lbs/in2  including  temperature  stresses.  1'he 
roadway  is  supported  over  a  considerable  portion  of  the  span 
length  by  means  of  a  reinforced  floor  carried  on  vertical  walls. 


HALF  SECTION  AT  PIER 
FIG.  109. 

The  viaduct  contains   eight  spans  of  the  dimensions  shown  in 
the  illustration. 

Figs.  108  and  109  illustrate  another  design  for  the  same  via- 
duct mentioned  in  the  preceding  paragraph.*    This  design  was 


*  Eng.  News,  Vol.  LVII,  1907,  p.  178. 


288  ARCHES.  [Cn.  VJII. 

submitted  by  Mr.  C.  A.  P.  Turner,  of  Minneapolis,  Minn.  In 
this  case  the  arch  is  composed  of  three  ribs  4  ft.  9  in.  square  at 
the  crown  and  7  ft.  3  in.  square  at  the  pier.  The  rib  reinforce- 
ment is  composed  of  longitudinal  rods  arranged  in  a  circle  and 
connected  at  frequent  intervals  by  bands  of  2J"X-f"  metal. 
The  stirrups  and  bent  bars  in  the  floor-beams  and  slabs  give  a 
very  effective  reinforcement. 


CHAPTER  IX. 

RETAINING-WALLS  AND  DAMS. 

187.  Advantages    of   Reinforced    Concrete.— Retaining- 
walls,  dams,  bridge  abutments,  and  the  like  constitute  a  class  of 
structures  in  which  the  outside  forces  acting  are  mainly  hori- 
zontal, and  in  which,  therefore,  the  question  of  stability  is 
largely  a  question  of  safety  against  overturning.    Where  ordi- 
nary masonry  is  used  in  these  structures  the  weight  of  the 
material  must  be  depended  upon  to  balance  the  overturning 
forces;  for  though  the  structure  be  anchored  to  the  foundation 
no  tensile  stresses  can  be  allowed  in  the  masonry.    As  a  con- 
sequence of  these  limitations  the  maximum  compressive  stresses 
in  such  structures  are  not  high,  except  in  extreme  cases,  so  that 
generally  the  dimensions  are  determined  by  the  weight  of  the 
material.     The  application  of  reinforced  concrete  in  such  cases 
enables  the  design  to  be  so  modified  as  to  utilize  the  weight  of 
the  material  to  be  retained  as  part  of  the  resisting  weight  and 
to  calculate  the  sections  to  develop  more  nearly  the  full  strength 
of  the  concrete.    A  very  considerable  gain  in  economy  therefore 
results. 

RETAINING-WALLS. 

188.  Method  of  Determining  Stability.— No  attempt  will 
be  made  here  to  present  the  various  mathematical  theories  of 
earth  pressure.    Unless  the  results  obtained  from  such  theories 
are  carefully  controlled  by  the   results  of   experience  they  are 
apt  to  be  very  misleading.     Probably  the  most  satisfactory  way 
to  design  a  reinforced  concrete  retainirig-wall,  as  regards  sta,- 
bility  against  overturning,  is  to  proportion  it  so  that  it  will  be, 

2S9 


290  RETAINING-WALLS  AND  DAMS.  [Cn.  IX. 

as  nearly  as  possible,  equivalent  to  a  solid  masonry  wall  of  such 
a  section  as  is  known  to  have  given  satisfactory  results  under 
the  given  conditions.  Rules  of  practice  as  to  solid  masonry 
walls  have  long  been  established.  They  represent  the  accu- 
mulated experience  of  many  engineers  and  are  based  upon  data 
obtained  from  many  failures  as  well  as  from  successful  designs. 
Until  experience  is  had  directly  with  the  reinforced  type  of  wall 
its  stability  may,  therefore,  well  be  determined  by  comparison 
with  the  older  form  of  construction.  The  analysis  given  here 
will  consequently  be  limited  to  a  convenient  method  of  com- 
parison of  the  two  types.  It  may  be  said  in  passing  that  good 
construction  requires  quite  as  much  attention  to  the  earth 
filling  itself  and  to  its  drainage  as  to  the  design  and  construc- 
tion of  the  wall. 

In  dimensioning  a  reinforced  concrete  wall  which  will  possess 
stability  equal  to  that  of  a  given  solid  wrall,  it  will  be  convenient 
to  determine  the  equivalent  fluid  pressure  under  which  the 
solid  wall  will  be  stable  and  then  apply  this  pressure  to  the 
reinforced  type  of  wall.  The  basis  of  the  calculation  of  this 
fluid  pressure  will  be  to  determine  the  weight  per  cubic  foot  of  a 
fluid  which  will  exert  such  a  pressure  against  the  solid  wall  as 
to  cause  the  resultant  of  all  forces  above  the  base  to  intersect 
the  base  at  the  edge  of  the  middle  third.  If,  then,  the  reinforced 
wall  be  designed  so  that  it  will  be  equally  stable  against  this 
pressure,  it  will  be  practically  equivalent  to  the  solid  wall. 

It  will  be  seen  that  this  method  is  very  simple  and  adapts 
itself  readily  to  the  utilization  of  present  rules  of  practice.  If 
desired,  the  theory  of  earth  pressure  may  of  course  be  directly 
applied  to  the  problem. 

189.  Equivalent  Fluid  Pressure  for  Ordinary  Masonry 
Walls.— Two  forms  of  wall  will  be  considered  (Fig.  110).  Form 
(a)  is  the  more  common  form  of  wall.  A  small  batter  is  usu- 
ally given  to  the  front  face,  and  the  back  face  is  sloped  in  an 
irregular  line,  the  width  of  the  top  being  as  narrow  as  circum- 
stances may  warrant.  Such  a  wall  will  be  stable  when  the 
width  of  the  base  is  made  from  one-third  to  one-half  the  height, 


§  189.] 


FLUID  PRESSURE  FOR  MASONRY  \VALLS. 


291 


four- tenths  being  a  common  rule  of  practice.  Form  (6)  is  used 
for  relatively  low  walls.  Its  width  may  be  a  little  less  than  that 
of  form  (a)  for  equal  stability.  While  the  calculations  here 
given  apply  only  to  the  two  forms  as  represented  in  Fig.  110, 
the  results  will  be  but  little  different  for  walls  similar  in  form 
but  which  vary  considerably  therefrom. 

Form  (a).— The  height  is  h  and  the  bottom  width  L  The 
batter  of  the  front  face  will  be 
taken  at  1:12,  and  the  top 
width  at  1/6  of  the  bottom 
width.  The  weight  of  the 
masonry  will  be  assumed  at 
150,  and  that  of  the  earth 
filling  at  100  lbs/ft3.  It  will 
be  assumed  that  the  fluid  pres- 
sure acts  against  a  vertical 
plane  FC]  the  stability  of  the  Pja  UQ 

entire  volume  to  the  left  of  this 

plane,  including  the  weight  of  the  earth,  will  be  determined. 
Let  Wi  denote  the  weight  of  masonry  per  lineal  foot,  and  W2 
the  weight  of  the  earth  filling  to  the  left  of  FC.  Let  P  denote 
the  resultant  fluid  pressure  acting  at  a  distance  J/i  above  the 
base.  Let  p  denote  the  weight  per  cubic  foot  of  such  fluid. 


Assume  that  the  resultant  pressure  due  to  the  weight  of  the 
wall  TFi,  the  weight  of  the  earth  W?,  and  the  pressure  P,  inter- 
sects the  base  at  the  edge  of  the  middle  third.  Equating  mo- 
ments about  this  point  we  derive  the  relation 


Form  (&).  —  In  this  form  the  only  forces  to  be  considered  are 
the  weight  Wt  and  the  pressure  P.  Equating  these  as  before 
there  results 


(2) 


292  RETAINING-WALLS  AND  DAMS.  [Cn.  IX. 

If  the  top  width  in  form  (a)  be  made  zero  the  effect  on  the 
result  would  be  to  change  the  coefficient  in  Eq.  (1)  from  132  to 
127,  thus  showing  that  a  considerable  variation  in  top  width 
has  little  effect  on  the  result. 

Substituting  various  values  of  l/h  in  Eqs.  (1)  and  (2)  we  have 
the  f  ollowing  values  of  p,  or  equivalent  fluid  weight,  under  which 
the  wall  is  stable  as  above  assumed. 

l/h  p 

1/3  14.71bs/ft3 

Form  (a)    |     4/10  21.1      " 

1/2  33.0      " 


Form  (6) 


1/4  9.4 

1/3  16.7 

4/10  24.0 


According  to  these  calculations  a  fluid  weight  of  20  to 
25  lbs/ft3  may  be  taken  as  a  basis  of  design  to  secure  sta- 
bility equivalent  to  the  ordinary  wall,  assuming  the  resultant 
pressures  to  cut  the  edge  of  the  middle  third  and  counting  the 
weight  of  earth  vertically  above  the  back  slope  as  part  of  the 
resisting  load.  It  is  to  be  noted  that  the  pressures  herein 
determined  are  not  necessarily  the  actual  earth  pressures;  the 
results  are  to  be  used  only  as  a  means  of  securing  stability  of 
reinforced  walls  approximately  equal  to  that 
of  solid  walls  of  known  proportions. 

190.  Stability    of   Reinforced   Concrete 
Walls.— Fig.  Ill  represents   in  outline  the 
usual  type  of  reinforced  wall.    It  consists  of 
a  vertical  wall  AE  attached  to  a  floor  DC. 
For  low  walls  the  upright  part  AE  may  act 
simply  as  a  cantilever;  and  likewise  the  parts 
EC  and  ED.    For  larger  walls  the  part  AE  is 
FIG.  ill.          tied  to  EC  at  intervals  by  back  walls  ACE  in 
the  form  of  narrow  transverse  walls  with  ten- 
sion reinforcement.    The  projecting  portion  ED  may  still  act 
as  a  cantilever,  or  it,  also,  mav  be  connected  to  the  vertical 


§  190.]  REINFORCED  CONCRETE  WALLS.  293 

wall  AE  by  means  of  buttresses.  In  either  case  the  earth 
pressures  act  in  essentially  the  same  manner  and  the  necessary 
width  of  base  is  found  in  the  same  way. 

Let  1=  width  of  base; 

x=  distance  from  toe  to  back  of  wall  AE; 

h=  height; 

p=  equivalent  fluid  weight  as  determined  in  Art.  189; 
11)2=  weight  of  earth  filling  per  cubic  foot; 
Wi  =  weight  of  masonry  per  lineal  foot; 
TF2=weight  per  lineal  foot  of  earth  above  the  floor  EC', 

a  =  lever-arm  of  Wi  about  point  F,  the  edge   of   the 
middle  third; 

P=  total  fluid  pressure  =  %ph2. 

Then  equating  moments  about  the  point  F  we  have 


,    .....    (3) 
or 

?i  .....     (4) 


If  the  wall  AE  is  placed  well  towards  the  front  the  moment 
of  the  masonry  will  be  small.  Neglecting  this  term  and  putting 
x=kl  we  may  solve  for  I,  getting 


This  is  a  minimum  for  k=-  J,  that  is,  for  x  =  $.    With  this 
value  of  k  we  have 


For  w 

J-.087Vp.fc  .......    (7) 

If,  for  example,  the  value  of  p  be  taken  at  21.1,  correspond- 
ing to  a  value  of  Z/fc  =  4/10  for  a  solid  wall,  the  value  of  I  is 


294 


RETAINING-WALLS  AND  DAMS. 


[Cn.  IX. 


equal  to  .Q87X^/21.IXh  =  Ah,  or  the  same  as  the  width  of  the 
solid  wall. 

As  it  may  be  desirable  to  use  a  smaller  or  larger  value  of  x 
than  JZ,  Table  No.  22  has  been  prepared  giving  the  values  of 
l/h  for  various  values  of  x/l  and  various  values  of  p.  An  ex- 
amination of  the  table  shows  plainly  that  the  length  of  the  pro- 
jection x  makes  very  little  difference  in  the  required  total 
length  of  base.  However,  with  x  made  very  small  or  very 
large  the  weight  of  the  wall  should  be  taken  into  account.  A 
further  fact  brought  out  by  the  table  and  by  the  table  of  Art. 
189  is  that  the  stability  of  the  reinforced  wall  is  about  the  same 
as  a  solid  wall  of  form  (a)  shown  in  Fig.  110  and  having  the 
same  base  length. 

TABLE  No.  22. 

PROPORTIONS  OF  REINFORCED-CONCRETE  RETAINING-WALLS. 

(See  Fig.  111.) 

VALUES  OF  l/h  FOR  DIFFERENT  VALUES  OF  p  AND  FOR  w2=100  (Eq.  (5)). 


Values  of  Equivalent  Fluid  Weight  p.     Pounds  per  Cubic  Foot. 

Values  of 

15 

20 

25 

33 

.5 

.35 

.40 

.45 

.51 

.33 

.34 

.39 

.43 

.50 

.25 

.34 

.39 

.44 

.50 

.20 

.34 

.40 

.44 

.51 

.15 

.35 

.40 

.45 

.52 

.10 

.36 

.41 

.46 

.53 

0 

.39 

.45 

.50 

.57 

The  resultant  forces  acting  upon  the  three  parts  of  the  wall 
AE,  DE,  and  EC  must  be  determined.  On  the  wall  AE  the 
force  may  be  taken  as  a  horizontal  force  equal  to  P,  =  Jp^2,  and 
applied  a  distance  $h  above  the  base.  The  resultant  force 
acting  on  any  length  h'  from  the  top  is  likewise  ^ph'2  and  ap- 
plied a  distance  %hf  below  the  top.  The  pressure  on  the  founda- 
tion will  equal  the  total  weight  TFi  +  TF2  and  will  be  applied  a 
distance  JZ  from  point  D.  The  average  unit  pressure  will  be 


§  191.]  DESIGN  OF  WALL.  295 

(Wi  +  W^/l,  and  the  maximum  pressure  at  D  will  be  twice 
this  value. 

The  upward  pressure  under  the  cantilever  DE  will  vary  from 

a  maximum  of  2 — —. — -  at  D  to  a  value  under  the  point  E  of 

2 — —j — ~X~T~'     This    is    a    "trapezoid"    of    pressure,    and 

where  x  is  large  the  centre  of  gravity  of  the  trapezoid  may  be 
found  and  the  resultant  applied  at  this  point.  Usually  it  will 
be  accurate  enough  to  assume  the  pressure  on  DE  as  uniformly 
distributed  at  an  average  value  and  applied  at  the  centre  of 
the  projection  outside  of  the  vertical  wall. 

The  upward  pressure  on  the  floor  EC  varies  from  the  value 
above  given  at  E,  to  zero  at  C.  It  varies  uni- 
formly between  these  points.  The  downward 
pressure  is  the  weight  of  the  earth  above  the 
floor,  =W2.  This  may  be  assumed  as  uniformly 
distributed  and  equal  to  V;2h  per  unit  area  at  all 
points.  The  total  downward  pressure  on  EC  will 
be  greater  than  the  upward  pressure  unless  xis 
very  small. 

191.  Design  of  Wall. — In  discussing  the  design  it  will  be 
necessary  to  consider  two  forms :  (1)  the  cantilever  wall  without 
back  tie-walls  as  in  Fig.  112,  and  the -wall  provided  with  such 
back  walls  as  in  Fig.  113. 

The  form  of  Fig.  112  is  adapted  to  heights  of  about  12  to 
18  feet.  For  high  walls  the  form  of  Fig.  113  will  be  more 
economical. 

Form    (a).  (Fig.    112.) — The    maximum    moment    in    the 

h    ph? 
upright  portion  AE  is  P^= ~ .    At  any  distance  h'  below 

the  top  the  moment  is  *?-.    Only  a  portion  of  the  reinforcing- 

rods  need  be  carried  up  the  full  height.    The  shear  at  the  bot- 

ph2 
torn  is  P***-sr.    This  will  be  very  small  and  will  require  no 


296  RETAINING-WALLS  AND  "DAMS.  [Cn.  IX. 

special  attention.  The  reinforcmg-rods  of  a  cantilever  beam 
have  their  maximum  stress  at  the  end  of  the  beam,  hence  special 
care  must  be  given  to  secure  an  effective  bond  or  anchorage. 
In  the  figure  the  vertical  rods  have  an  insufficient  length  below 
the  point  of  maximum  moment  to  develop  their  full  strength, 
and  therefore  they  should  be  anchored  in  a  substantial  man- 
ner. This  may  be  done  by  screw-ends  and  nuts,  or  by  loop- 
ing the  rods  around  anchor-bars  near  the  bottom  of  the  floor 
DC. 

The  cantilever  DE  must  be  treated  in  the  same  manner  as 
the  upright  cantilever.  The  .pressures  will  be  much  heavier 
and  the  shear  and  bond  stress  may  need  attention.  The  rein- 
forcement should  extend  far  enough  beyond  E  for  bond  strength. 

The  cantilever  EC  is  acted  upon  by  an  upward  and  a  down- 
ward force  as  shown  in  the  figure.  The  maximum  moment  will 
be  at  E  and  will  be  negative.  It  is  provided  for  by  reinforce- 
ment as  shown. 

To  secure  maximum  economy  each  one  of  the  cantilevers 
may  be  tapered  towards  the  end  to  a  minimum  practicable 
thickness.  The  bending  moments  at  various  sections  in  a 
cantilever  beam  uniformly  loaded  vary  as  the  squares  of  the 
distances  from  the  free  end.  The  resisting  moments  vary  ap- 
proximately as  the  squares  of  the  depths  of  the  beam.  Hence  a 
team  tapering  uniformly  to  zero  depth  at  the  end  would  be  of 
the  necessary  depth  at  all  points.  The  moments  in  the  vertical 
beam  AE  vary  as  the  cubes  of  the  distances  below  the  top,  so 
that  a  straight  taper  will  in  this  case  give  a  beam  whose  weakest 
point  will  be  at  the  bottom.  At  the  top  point  A  some  form  of 
coping  is  usually  added,  of  a  width  according  to  the  require- 
ments of  the  case. 

To  prevent  unsightly  cracks  a  certain  amount  of  longitudinal 
reinforcement  is  necessary.  The  amount  required  per  square 
foot  of  cross-section  will  be  less  the  heavier  the  wall,  as  tem- 
perature changes  will  be  less  in  such  a  wall.  On  the  basis  of 
the  discussion  in  Chap.  V,  Art.  142,  the  percentage  required 
may  be  placed  at  about  0.4%  as  a  maximum  for  thin  walls,  to 


§  191.] 


DESIGN  OF  WALL. 


297 


perhaps  one-half    of  this  for  heavy  walls.     High  elastic-limit 
material  is  advantageous  for  this  purpose. 

Form  (b).  (Fig.  113.) — So  far  as  the  external  pressures  are 
concerned  they  have  been  explained  in  Art.  189,  and  are  prac- 
tically the  same  as  in  the  previous  case  considered.  The  loads 
or  pressures  on  the  concrete  are,  however,  carried  quite  differ- 
ently. The  toe  DE  is  the  same  as  in  form  (a)  and  reinforced 
in  the  same  way.  The  pressure  against  the  longitudinal  wall  AE  is 
carried  laterally  for  the  most  part  and  given  over  to  the  inclined 


(a) 
CROSS-SECTION 


I 

ii 

1  1 
1  1 
I 

; 

! 



jj 



j- 



,  

ri- 



+  ^ 



i 

i 

1 

i 

1 

i 

I 

h 

i 

it 

I 

H 

I 

ii 

i 

i 

! 

! 

I 

i* 

Ii 

h 

(b) 

h' 

ELEVATIOf 

4 

JJL. 


PLAN 


FIG.  113. 


back  walls.  The  wall  AE  must  therefore  be  designed  as  a  slab 
supported  along  the  lines  AE  and  A'E'  (Fig.  113  (6)),  and  sub- 
jected to  a  pressure  per  square  foot  at  any  point  a  distance  h' 
below  the  top  equal  to  ph'.  Near  the  bottom,  the  load  on  AE 
is  transmitted  more  or  less  to  the  floor  EC.  The  wall  should 
therefore  be  bonded  to  the  floor  with  a  small  amount  of  vertical 
reinforcement,  which  may  well  extend  to  the  top  to  prevent 
cracks,  although  under  ordinary  conditions  the  wall  AE  is  under 
some  vertical  pressure. 

The  floor  EC  is  subjected  to  both  upward  and  downward 
pressures,  the  latter  exceeding  the  former  towards  the  end  C, 


298  RETAINING-WALLS  AND   DAMS.  [On  IX. 

and  possibly  throughout,  as  previously  explained.  This  floor 
is  supported  by  the  back  wall  AEC  and  is  therefore  reinforced 
longitudinally  as  a  floor-slab  in  accordance  with  the  resultant 
pressure  at  any  point.  Here,  again,  it  is  well  to  bond  the  floor 
to  the  wall  AE  by  extendbg  the  transverse  reinforcement  of 
the  toe  DE  into  the  portion  EC. 

The  back  w^all  ACE  acts  as  a  cantilever  beam  anchored  to 
the  floor.  It  is  also  a  T-beam,  the  flange  being  the  longi- 
tudinal wall  AE.  The  tension  along  the  edge  AC  is  carried  by 
rods  near  this  edge,  wrhose  stress  at  any  point  is  found  with 
sufficient  accuracy  by  an  equation  of  moments  taken  about  the 
center  of  the  front  wall.  The  maximum  stress  will  be  at  the 
bottom.,  if  the  wall  is  made  with  a  straight  profile.  At  the 
connection  of  the  wall  AEC  to  the  floor,  it  is  to  be  noted  that 
the  floor  load  is  transferred  to  the  wall  along  the  line  EC, 
but  mainly  near  the  end  C.  The  main  tension-rods  in  AC 
should  therefore  be  distributed  somewhat  at  their  lower  ends 
and  well  anchored  to  the  reinforcing-rods  of  the  floor  EC.  A 
few  additional  vertical  rods  should  also  be  put  in  to  insure 
thorough  bonding  of  floor  to  wall.  These  will  also  carry  a  part 
of  the  tension  in  the  back  wall,  but  will  not  be  as  efficient  as 
A  the  rods  nearer  the  outside  edge.  It  is  desira- 

ble, likewise,  to  bond  the  vertical  wall  AE  to  the 
back  wall  with  short  horizontal  rods  as  shown. 
The  slabs  formed  by  the  walls  AE  and  the 
floor  EC  are  continuous  over  supports,  and  if 
the  span  is  long  should  be  provided  with  some 
reinforcement  for  negative  moments  at  these 
supports. 

Fig.  114  shows  some  additional  features  of 
design  w^hich  have  been  used.    A  longitudinal 
c  beam  is  built  at  C  and  the  floor  is  thus  supported 
on  all  four  edges.     The   main  rods  along  AC 
are  then  anchored  into  the  beam. 

A  horizontal  beam  may  also  be  made  of  the  coping  at  A, 
thus  giving  some  support  to  the  wall  AB  along  its  upper  edge. 


§  192.] 


ILLUSTRATIIV  EXAMPLES. 


299 


A  projection  may  be  necessary  at  the  toe  7),  or  elsewhere,  in 
order  to  increase  the  resistance  against  forward  sliding.  The 
beam  C  aids  in  this  respect. 

192.  Illustrative  Examples.— Fig.  115  shows  the  form  of 
retaining-wall  used  on  the  Great   Northern  R.R.  at  Seattle, 


SECTION 


-•J\ilm^-----'A\i\m.J 


ELEVATION 
6"Weepers 


«rt 


dirt's 


PLAN 


FIG.  115. — Retaining-wall,  Great  Northern  Railway. 

Wash.*  This  is  a  good  illustration  of  the  second  type  above 
discussed.  An  estimate  by  Mr.  C.  F.  Graff  of  the  amounts  of 
material  per  lineal  foot  required  in  reinforced  and  plain  con- 


*  Eng.  News,  Vol.  LIII,  1905,  p.  262. 


300 


RETAINING-WALLS  AND  DAMS. 


[Cn.  IX. 


§  193.]        RETAINING-WALLS  SUPPORTED  AT  THE  TOP. 


301 


crete  walls,  made  in  connection  with  the  design  of  Fig.  115,  gave 
the  following  results: 


Amount  of  Concrete  per  Lineal  Foot. 

Height  of  Wall,  Feet. 

Saving:   Per  Cent  of 
Reinforced  Wall. 

Plain  Wall, 

Reinforced  Wall, 

Cubic  Feet. 

Cubic  Feet. 

40 

396.4 

218 

45 

30 

226 

127.8 

43.3 

20 

110 

69.9 

36.4 

10 

44 

34.9 

20.4 

The  steel  was  included  by  adding  its  concrete  equivalent. 

Fig.  116  illustrates  a  standard  form  of  abutment  used  by  the 
Wabash  R.R.  Co.* 

193.  Retaining-walls  Supported  at  the  Top.— Frequently  a 
retainiug-wall  may  be  supported  at  the  top.  In  such  a  case  it  is 
designed  as  a  simple  beam  supported  at  the  top 
and  bottom;  or  vertical  ribs  or  beams  may  thus 
be  calculated  and  the  slab  reinforced  horizontally 
and  supported  by  these  ribs. 

A  wall  A B  (Fig.  117)  acted  upon  by  a  pressure 
uniformly  varying  from  zero  at  the  top  to  a  maxi- 
mum at  the  bottom  will  be  subjected  to  a  bending 
moment  whose  maximum  value  will  be  determined. 
Let  the  pressure  be  that  due  to  a  fluid  weighing 
p  lbs/ft3.  Then  P  =  ip/i2,  R1=%P  =  %ph2,  R2=-^ph2. 

The  bending  moment  M  at  a  distance  x  below 
A  =  R&-  &x* = p/6  (h2x  -  z3) . 

This  is  a  maximum  for  x  =  W  J  =  .58/1.    The  maximum  moment 
is  then 

A/  =  .064p#» (8) 

If  the  pressure  is  water  pressure,  as  in  a  reservoir,  the  value 
of  the  maximum  moment  becomes  equal  to 

M =4fc3, (9) 

where  the  units  are  the  foot  and' pound.    For  an  earth  retaining- 
wall  with  p  =  20,  then  M  =  13h*,  etc. 

*  Ry.  Rev.,  Vol.  XLV,  1905,  p.  523. 


B 

FIG.  117. 


302 


RETAINING- WALLS  AND  DAMS. 


[On.  IX. 


DAMS. 

194.  The  dam  is  a  form  of  retaining-wall,  but  is  subject  to 
somewhat  different  conditions  as  to  pressures.  For  this  case  a 
form  of  wall  as  shown  in  Fig.  118  is  poorly  adapted,  owing  to 
the  fact  that  the  water  pressure  will  probably  penetrate  beneath 
the  floor  DC  and  exert  an  upward  force  nearly  equal  to  the 
downward  pressure,  thus  destroying  the  usefulness  of  the  floor 
EC.  To  obviate  these  objections  the  wall  AE  must  be  brought 
back  to  the  point  C.  Increased  stability  will  then  be  secured 
by  making  it  inclined.  In  this  position  it  will  naturally  be 
supported  by  transverse  walls  or  buttresses,  resting  on  a  floor 
DC,  or  directly  on  the  foundation  material,  as  shown  in  Fig. 
119.  The  water  pressure  on  the  floor  may  then  be  relieved  by 


D          E 

FIG.  118. 


FIG.  119. 


drain-openings  allowing  free  exit  for  seepage-water.  Thus 
built  it  forms  a  stable  and  efficient  type  of  dam.  Its  design  as 
to  stresses  and  sections  is  simple  and  obvious.  The  wall  or 
floor  AC  may  be  supported  directly  on  the  cross-walls  and  re- 
inforced with  longitudinal  rods,  or  longitudinal  beams  may  be 
used  as  shown  and  the  slab  supported  on  these.  The  pressure 
on  the  foundation  is  determined  by  considering  the  resultant 
of  water  pressure  and  weight  of  dam.  The  buttresses  or  cross- 
walls  are  subjected  only  to  compressive  stresses.  Ample  longi- 
tudinal reinforcement  should  be  provided  to  thoroughly  bind 
the  structure  together.  Dams  are  often  subjected  to  dynamic 
loads  as  well  as  static  pressures,  and  sections  must  be  provided 
more  liberally  than  in  many  other  structures. 


§  194.] 


DAMS. 


303 


The  form  shown  in  Fig.  119  is  not  suited  to  act  as  a  spillway 
except  for  low  falls.  For  a  spillway  the  down-stream  edge  of 
the  buttresses  is  also  covered  with  a  floor  which  may  be  curved 
in  the  usual  manner. 


Crest 


Vent 


FIG.  120.— Dam  at  Schuylerville,  N.  Y. 

Fig.  120  illustrates  a  dam  of  this  type  built  at  Schuylerville, 
N.  Y.,  by  the  Ambursen  Hydraulic  Construction  Co.*  A  foot- 
way is  provided  for  in  the  interior.  The  design  as  to  strength 
is  obvious. 

*  Eng.  News,  Vol.  LIII,  p.  448. 


CHAPTER  X. 

MISCELLANEOUS     STRUCTUKES. 
GIRDER   BRIDGES   AND   CULVERTS. 

195.  For  short  spans,  the  girder  bridge  or  box  culvert  is 
likely  to  be  a  more  economical  form  than  the  arch,  owing  to 
the  less  rigid  requirements  for  foundations  and  abutments. 
For  purposes  of  analysis  this  type  of  structure  may  be  divided 
roughly  into  three  classes:  (1)  Simple  spans  in  which  the 
girder  rests  upon  independent  abutments  or  piers;  (2)  con- 
crete trestles  or  bridges  in  which  the  girders,  abutments  and 
piers  form  a  monolithic  structure;  and  (3)  pipe  culverts  and 
box  culverts  built  as  square  or  rectangular  pipes. 

196.  The  Simple  Beam  Bridge.  —  These  are  designed  in  the 
same  manner  as  any  other  concrete  floor.  Spans  up  to  20  to 
30  feet  may  well  be  made  as  a  simple  slab  or  uniform  thickness 
spanning  the  opening.  For  railroad  structures  the  loads  are 
relatively  so  large  that  shearing  stresses  will  usually  require 
careful  attention.  For  longer  spans  a  gain  in  economy  will 
result  by  the  use  of  main  horizontal  girders  of  relatively  great 
depth,  with  a  floor  supported  by  the  girders  and  reinforced 
transversely.  The  bridge  may  be  made  either  a  "  through  "  or 
"  deck  "  girder,  according  to  the  requirements  of  the  case,  the 
latter  being  the  more  economical.  Floors  of  reinforced  con- 
crete are  also  used  for  steel  truss  and  girder  bridges  to  a  con- 
siderable extent  where  a  solid  floor  is  desired.  The  details  are 
arranged  in  a  variety  of  ways,  but  the  calculation  and  design 
of  the  reinforcement  to  meet  the  given  conditions  require  no 
special  consideration.  The  proper  allowance  for  impact  is  an 

304 


§  197.]  CONCRETE    TRESTLES.  305 

important  point  in  this  connection.  Durability  is  an  important 
factor  favorable  to  the  use  of  reinforced  concrete  for  bridge 
floors. 

197.  Concrete  Trestles.  —  Where  several  short  spans  are 
required  and  concrete  is  used  for  both  the  girders  and  the 
piers,  the  latter  may  usually  be  made  of  comparatively  small 
cross-section,  —  much  smaller  than  possible  if  ordinary  masonry 
be  used.  The  structure  then  approaches  the  ordinary  floor 
and  column  construction  in  the  relations  of  its  parts.  The 
piers,  if  lightly  loaded,  may  consist  merely  of  two  or  more 
columns  connected  by  a  suitable  portal.  In  some  extreme 
cases  designs  have  been  carried  out  in  which  the  supporting 
piers  or  towers  have  been  arranged  in  a  manner  similar  to  a 
steel  trestle,  even  to  the  diagonal  bracing.  It  would  seem, 
however,  that  the  treatment  of  concrete  should  be  on  some- 
what different  lines  than  is  best  suited  to  such  a  material  as 
steel,  and  that  structural  forms  in  concrete  should  be  somewhat 
massive  and  limited  in  general  to  the  beam  and  the  compression 
member. 

Where  the  piers  are  made  small,  as  here  assumed,  they  must 
be  built  rigidly  in  connection  with  the  girders  of  one  or  more 
spans,  as  are  the  columns  in  a  building.  The  girders  must  be 
designed  with  proper  reference  to  their  continuity,  and  the  piers 
must  be  able  to  resist  a  certain  amount  of  bending  moment. 
This  moment  can  be  estimated  in  the  manner  suggested  in 
Chapter  VII,  Art.  167. 

As  an  example,  let  Fig.  121  represent  a  concrete  trestle  of 
monolithic  construction.  The  girders  are  continuous  and  the 


j 


FIG.  121. 


piers  are  rigidly  attached  to  them.     The  greatest  moment  in 
the  pier  BF  will  occur  when  one  of  the  spans  AB  or  EC  is  loaded. 


306  MISCELLANEOUS   STRUCTURES.  [Cn.  X. 

Suppose  EC  be  loaded.  Then  calculate  the  negative  moment 
at  B,  assuming  EG  to  be  fixed  at  the  ends.  This  moment  will 
be  equal  to  -  A  pi2,  where  p  =  load  per  foot  and  /  =  span 
length.  Now  this  moment  is  distributed  at  the  joint  B  among 
the  three  members  A  B,  BF,  and  BC  in  proportion  to  the 
value  of  I/I  for  the  three  members,  the  length  I  being  taken  as 
Ihe  estimated  length  to  the  point  of  inflection  in  each  case 
(the  full  length  of  BF).  This  will  determine  approximately 
the  moment  in  BF.  The  maximum  negative  moment  in  BO 
and  A  B  will  occur  when  both  spans  are  loaded  and  will  be 
approximately  equal  to  TV  pi2-  (See  Chapter  VII,  Art.  155.) 
The  end  piers  or  abutments  must  be  designed  also  as  retaining 
walls. 

198.  Pipe  and  Box  Culverts.  —  For  small  openings  the  mono- 
lithic pipe  or  box  form  is  very  advantageous.     This  form  of 
structure  is  a  complete  opening  in  itself  and  so  long  as  intact 
will   do   good   service.    Considerable    settlement,   as   a   whole, 
may  be  permissible,  and  hence  solid  foundations  may  not  be 
needed. 

The  cross-section  may  be  circular,  elliptical  or  rectangular. 
Theoretically,  the  elliptical  form  is  the  best  as  corresponding 
more  nearly  to  the  requirements  for  resisting  the  earth  pressure. 
The  circular  is  practically  as  good  for  small  openings,  while 
for  large  openings  the  rectangular  form  will  often  be  the  best 
on  account  of  its  simplicity  and  the  lesser  head  room  required. 
Where  the  culvert  is  manufactured  at  a  shop  and  transported 
to  the  site,  the  circular  or  elliptical  forms  will  usually  be  the 
most  advantageous.  As  the  loads  coming  upon  such  structures 
are  not  accurately  known  an  exact  analysis  of  the  stresses  is 
impossible,  but  the  results  obtained  for  certain  simple  cases 
will  be  useful  as  a  guide  to  the  judgment.  The  general  method 
of  analysis  employed  in  Chapter  VIII  has  been  used.  The 
details  of  the  analysis  will  be  omitted. 

199.  The    Circular    Culvert.  -  -  Two     cases     have     been 
analyzed;    (1)  for  a  uniform  load,  and  (2)  for  a  concentrated 
load. 


199.] 


PIPE    AND    BOX   CULVERTS. 


307 


pd 


Case  I;  Uniform  load.  (Fig.  122.)  It  is  assumed  that  the 
pressure  on  the  pipe  is  exerted  in  parallel  lines  (as  downward 
and  upward)  and  is  uniformly  distributed  with 
respect  to  a  plane  perpendicular  to  the  direction 
of  the  pressure.  JiU'UlU 

Let  d  =  diameter  of  pipe; 

p  =  pressure  per  unit  area  as  measured  per- 
pendicularly to  the  pressure; 
M  =  bending  moment  in  pipe  in  a  length  of 
one  unit; 


mttttjt 


Then  the  following  equations  result. 


Mc  =  Md  =  - 


FIG.  122. 


(1) 

(2) 


If  the  lateral  pressure,  measured  in  a  similar  way;  be  pf  per 
unit  area,  then  the  moments  due  to  this  pressure  will  be 


and 


Mc  =  Md  =  A  p'd 


(3) 
(4) 


For  equal  horizontal  and  vertical  forces  (equivalent  to  a  uni- 
form radial  pressure),  the  moments  at  all  points  are  zero. 
Usually  the  lateral  pressure  will  be  much  less  than  the  vertical 
pressure;  probably  not  more  than  one-fourth  or  one-fifth  as 
much.  Assuming  a  ratio  of  one-fourth,  the  resulting  total 
bending  moments  at  the  points  a,  b,  c,  d,  will  be  A  pd 2, 

positive  at  the  top  and  bottom  and  negative  at 

the  sides. 

Case  II;  Concentrated  loads  at  opposite  points 

(Fig.  123). 
In  this  case  the  moments  are 


FIG.  123. 


Ma  =  Mb  =  .16  Pd (5) 

Mc  =  Md  =  -  .09  Pd (6) 


308 


MISCELLANEOUS    STRUCTURES. 


[Cn.  X. 


200.       The   Rectangular   Culvert.  —  Case   I;    Uniform    loads 
(Fig.  124). 

Let  ^  =  width  of  culvert; 
12  =  height  of  culvert; 

7t  =  moment  of  inertia  of  top  and  bottom,  assumed  as  equal  ; 
72  =  moment  of  inertia  of  sides; 
p  =  vertical  load  and  foundation  reaction  per  unit  area. 


Then 


M     -M     - 


The  moments  at  e  and  /  are  equal  to  M  c. 


(7) 
(8) 


4UUUHHJHU 


T 

Jc 

f 

o 

I 

in 

d 

1 

b 

J. 

J 

*- -£l 


tmttttttmtt 


FIG.  124. 

For  a  square  culvert  with  uniform  section  Ma  =  TV  pP  and 
Mc  =  —  &pl\ 

For  equal  vertical  and  lateral  loads  the  moments  in  the  square 
culvert  become  Ma  =  Mc  =  +  &  pP  and  Me  =  -  A  pP  as  in 
a  beam  with  fixed  ends. 

Case  II;  Concentrated  loads.     (Fig.  125.) 

For  vertical  loads  applied  centrally, 


, 

4 


t  + 

Mc  =  Md  =  Ma  -  \  PI, 


,g) 

(10) 


§  201.]  PIPE   AND   BOX    CULVERTS.  309 

For  the  square  form,  Ma  =  tV  Plt  and  Mc  =  --  TV  P^  ;  and 
for  equal  lateral  and  vertical  forces  Ma  =  Mc  =  ^  Pl^  and 
MP  =  —  i  PL  as  for  fixed  beams. 

e  o  1 

201.  Arrangement  of  Reinforcement. — The  bending  moments 
here  determined  are  based  on  the  assumption  that  the  entire 
section  is  reinforced  so  as  to  act  as  a  monolithic  structure. 
This  of  course  requires  proper  reinforcement  for  negative  as 
well  as  positive  moments. 

In  the  circular  form  a  wire  mesh  is  convenient,  especially 
for  small  diameters.  A  single  mesh  will  be  sufficient,  placed 
near  the  intrados  at  top  and  bottom  and  near  the  extrados 
at  the  sides,  crossing  the  central  axis  at  about  the  quarter 
point. 

In  the  rectangular  form,  if  reinforcement  for  negative  moments 
at  the  corners  is  omitted,  then  the  four  sides  will  act  as  simple 
beams,  the  concrete  cracking  more  or  less  on  the  outside  near 
the  corners. 

Longitudinal  reinforcement  should  be  provided  to  some 
extent.  Where  foundations  are  good  a  very  small  amount  will 
be  sufficient,  but  if  settlement  is  likely  to  occur  the  longitudinal 
reinforcement  becomes  of  much  importance.  The  entire  cul- 
vert will  act  as  a  beam  subjected  in  the  main  to  positive  bend- 
ing moments.  Most  of  the  reinforcement  should  therefore  be 
placed  along  the  bottom  of  the  culvert. 

202.  Illustrative  Examples.  —  Fig.   126  illustrates  a  simple 
beam   bridge   or   "  trestle  "   on  the   Chicago,   Burlington  and 
Quincy  R.R.*    The  girder  consists  of  a  slab  twenty-four  inches 
in  thickness,  reinforced  as  shown  in  the  illustration.    The  piers 
are  separate  structures. 

Fig.  127  represents  a  concrete  highway  bridge  as  an  over- 
head crossing  of  the  Big  Four  R.R.  This  design  illustrates 
the  deep  girder  with  floor-slab  reinforced  transversely,  and  also 
the  "  trestle  "  in  which  the  piers  are  columns  built  as  one  piece 
With  the  girders,  f 

*  R.  R.  Gaz.,  Vol.  XL,  1906,  p.  713. 
t  R.  R.  Gaz.  Vol.  XL,  1906,  p.  497. 


810 


MISCELLANEOUS   STRUCTURES. 


[Cn.  X. 


wWii-.~--w-y-! 

33      a^  =,   «\  *•        «„          M  !;•• '.;•-;  o 


203.] 


ILLUSTRATIVE    EXAMPLES. 


311 


Fig.  128  illustrates  a  standard  design  for  a  monolithic  box 
culvert.     It   is   not   reinforced    for   negative    moment   at   the 


CROSS  SECTION 


FIG.  128. 

corners.  This  form  of  construction  is  applicable  to  many  other 
structures  as  subways,  tunnel  linings,  etc.  No  special  con- 
sideration of  these  various  applications  of  the  reinforced  beam 
is  required  in  this  place.  A  clear  understanding  of  the  general 
principles  of  reinforced  concrete  design  will  enable  the  details 
to  be  suitably  modified  to  meet  the  conditions  of  the  case. 

CONDUITS  AND  PIPE  LINES. 

203.   For  conduits  not  under  pressure,  large  sewers  and  the 
like,  reinforced  concrete  lends  itself  to  convenient  and  economi- 


«**  #  10  Exp.  Metal 


FIG.  129. 


cal  construction.  As  to  the  analysis  and  design,  these  struc- 
tures are  only  special  cases  of  the  monolithic  pipe  or  box 
discussed  in  preceding  articles.  The  character  of  the  foundation 


312 


MISCELLANEOUS    STRUCTURES. 


[Cn.  X. 


and  convenience  in  construction  will  lead  to  various  modifi- 
cations of  design. 

Fig.  129  is  a  typical  cross-section  of  a  large  sewer  for  Harris- 
burg,  Pa.  A  mesh  of  expanded  metal  is  used  for  reinforce- 
ment, arranged  to  resist  positive  moments  excepting  at  bottom 
and  corners. 

Fig.  130  illustrates  a  large  conduit  of  the  Jersey  City  Water 
Supply.  This  section  is  employed  where  the  bottom  is  soft, 


^Longitudinal  )/TwisteJ 
'Steel  Rods  2 'apart 

%'Twisted  Steel  Rods  l'  apart 


Ends  LappedJ.' 
and  Wired 


Mesh*10  Expanded  Metal 


FIG  130. 


special  reinforcement  being  used  in  the  invert.  The  position 
of  the  reinforcement  to  carry  positive  moments  at  crown  and 
negative  moments  at  sides  should  be  noted. 

Reinforced  concrete  has  also  been  used  to  some  extent  for 
pipes  under  pressure,  but  it  is  very  difficult  to  secure  impervious- 
ness  under  heads  of  considerable  '  magnitude.  In  pressure 
pipes  the  tensile  stress  is  entirely  taken  by  the  steel,  the  concrete 
furnishing  merely  the  impervious  layer  and  resisting  bending 
due  to  earth  loading. 

TANKS,  RESERVOIRS,  BINS,  ETC. 

204.  For  covered  reservoirs  reinforced  concrete  is  very  well 
adapted.  The  rectangular  form  with  flat  cover  is  usually  the 
most  convenient;  its  design  involves  the  same  features  as  build- 


§204.]  CONDUITS   AND   PIPE   LINES  313 

ing  design  with  the  additional  one  of  imperviousness.  Elevated 
towers  and  tanks  may  also  be  made  of  concrete,  but  high 
pressures  are  difficult  to  deal  with. 

Bins  and  coal  pockets  are  structures  for  which  concrete  is 
well  adapted.  For  the  storage  of  coal  unprotected  steel  is  not 
durable,  but  reinforced  concrete  furnishes  an  almost  ideal 
material,  lending  itself  readily  to  the  necessary  form  for  strength 
and  furnishing  the  desired  durability. 

Tall  chimneys  constitute  also  a  type  of  structure  for  which 
reinforced  concrete  is  well  suited,  and  several  good  examples 
exist  of  these  structures.  They  are  built  monolithic,  usually 
in  sections  of  a  size  that  can  be  completed  in  one  working  day. 
The  entire  structure  is  designed  as  a  cantilever  beam  subjected 
to  bending  moments  from  the  wind  pressure  and  direct  com- 
pression due  to  its  own  weight.  At  any  section  the  stresses 
would  be  found  as  explained  for  the  case  of  flexure  and  com- 
pression in  Chapter  V.  Vertical  reinforcement  provides  for 
these  stresses,  while  circumferential  reinforcement  is  provided 
sufficient  in  amount  to  thoroughly  bond  the  structure  together. 
Working  stresses  similar  to  those  used  in  other  structures  may 
be  employed  if  the  work  is  carefully  executed.  Foundations 
are  designed  as  one  mass  and  proportioned  with  reference  to  the 
allowed  pressure  on  the  earth  under  the  maximum  overturning 
moment. 

Reinforced  concrete  is  advantageously  used  in  other  minor 
forms  of  structures  and  structural  elements.  For  further 
illustrations  the  reader  is  referred  to  the  larger  works  on  the 
subject  and  especially  to  the  American  works  of  Messrs.  Buell 
and  Hill  and  Mr.  Reid. 


INDEX. 


Adhesion,  33 

of  concrete  and  reinforcing  bars,  33 

tests  of,  34 

working  values  of,  172 
Analysis  of  arches,  265-282 
Arches, 

advantages  of  reinforced,  6,  263 

analysis  of,  265-282 

deflection  of,  274 

examples  of,  282 

methods  of  reinforcing,  264 

stresses  in,  266,  281 

temperature  stresses  in,  272 

unsymmetrical,  274 

Bars, 

forms  of,  30 
spacing  of,  174 

Beams, 

advantages  of  reinforced  concrete 

for,  5 

applicability  of  theory  of,  131,  169 
compression  reinforcement  in,  147 
continuous,  237,  240,  250 
depth  of,  177 
diagonal  tension  failures    of,  131, 

138 

diagrams  for  simple,  213-217 
double  reinforced,  84 
double  reinforement  of,  147 
economical  proportions  of,  175 
flexure  and  direct  stress  in,  90,  204 
formulas  for  (see  Formulas) 
methods  of  failure  of,  111 
shear  failures  of,  131,  138 
shear  reinforcement  for,  133 
shearing  stresses  in,  100,  133,  207 
T-beams  (see  T-beams) 
tension  failures  of,  119 
tests  of,  120,  138 
variation  of  stress  in,  43 
web  reinforcement  in,  133,  173 
working  stresses  in,  170 


Bins,  312 
Bond,  33 

mechanical,  35 

tests  of,  34 
Bond  stress,  105,  207 

working  values  of,  172 
Bridges  (see  Arches  and  Girders) 
Broken  stone,  general  requirements,  9 
Building  construction,  237 

advantages  of  reinforced  concrete 
in,  6 

Cement,  general  requirements,  9 
Chimneys,  advantages  of  reinforced 

concrete  for,  7 

Cinder  concrete,  properties  of,  29 
Coefficient  of  expansion  of 

concrete,  26 

steel,  33 
Columns, 

advantages  of  reinforced  concrete 
for,  5 

details  of,  254 

diagram  for,  221 

eccentric  loads  on,  253 

economy  of  reinforced  concrete  for, 
188 

examples  of  design  of,  254 

formulas  for,  107, 110 

hooped,  110,  159,  191 

notation  for,  106,  208 

reinforcement  of,  106 

steel  for,  189 

tests  of  plain  concrete,  151 

tests  of  reinforced,  153,  159 

working  stresses  for,  185 
Concrete, 

cinder,  29 

coefficient  of  expansion  of,  26 

consistency  of,  10,  13 

crushing  strength  of,  11 

in  beams,  129 

elastic  limit  of,  25 

315 


316 


INDEX. 


Concrete, 

elastic  properties  of,  19 

elongation  of,  37 

expansion  of.  26,  27 

general  requirements,  8 

modulus  of  elasticity  of,  20,  36 

proportions  of  ingredients  of,  10 

temperature  stresses  in,  40 

tensile  strength  of,  15 

transverse  strength  of,  16 

shearing  strength  of,  17 

shrinkage  of,  27 

shrinkage  stresses  in,  40 

stress-strain  curves  of,  26 

variation  in,  9 

weight  of,  29 

working  stresses  for,  170 
Conduits,  311 

advantages  of  reinforced  concrete 

for,  7      . 

Continuous  beams,  237,  240,  250 
Culverts, 

advantages  of  reinforced  concrete 
for,  6 

examples  of,  309 

reinforcement  for,  309 

stresses  in,  306 

Dams,  302 

advantages  of  reinforced  coecrete 

for,  6 
Diagrams  for 

circular  slabs,  222,  223 

columns,  221 

double  reinforcement,  218 

flexure  and  direct  stress,  219,  220 

simple  beams,  213-217 
Double  reinforcement,  diagram  for, 

218 
Double  reinforcement  of 

beams,  84,  147 

T-beams,  84 

Eccentric  column  loads,  253 

Factor  of  safety,  166 

Fireproofing,  reinforced  concrete  for, 

193 
Flexure  and  direct  stress,  90,  204 

diagram  for,  219,  220 
Floors,  237 

design  of,  244 

examples  of,  254 
Floor-beams, 

arrangement  of,  248 

continuous,  237 

loads  on,  249 


Floor-slabs, 

continuity  of,  240 

design  of,  244 

reinforced  in  two  directions,  241 

shrinkage  reinforcement  for,  244 

tables  for,  227-236 
Footings,  260 
Formulas  for 

beams,  compared,  77 

beams,  double-reinforced,  84,  202 

beams,  Talbot's,  69 

beams,  ultimate  loads,  62,  65,  199 

beams,  variety  of,  52 

beams,  working  loads,  55,  65,  169, 
197 

columns,  107,  110 

T-beams,  79,  201 
Frictional  resistance  of  bars,  35 

Girders  (see  Beams) 
Girder  bridges  304 
Gravel,  general  requirements,  9 

Historical  sketch,  1 
Live  load,  effect  of,  167 

Mechanical  bond,  35 
Melan  system,  1 
Modulus  of  elasticity  of 

concrete,  20,  36 

steel,  32 

Monier  system,  1 
Mushroom  system,  259 

Notation  for 
beams,  55,  197 

beams,  double-reinforced,  84,  202 
columns,  107,  208 
flexure  and  direct  stress,  91,  205 
T-beams,  79,  201 

Piles,  use  of  reinforced  concrete  for,  7 
Pipes,  306,  311 
Plates,  circular, 

diagrams  for,  222,  223 

stresses  in,  210 

Railroad  ties,  use  of  reinforced  con- 
crete for,  7 
Ransome  system,  2 
Reinforced  concrete, 

advantages  of,  4 

durability  of,  191 

fire  resisting  qualities  of,  193 

history  of,  1 

repeated  load  tests  of,  1 64 

shrinkage  stresses  in,  40 


INDEX. 


317 


Reinforced  concrete, 

temperature  stresses  in,  40 

use  of,  4 

working  stresses  for,  166,  211 
Reservoirs,  312 

advantages  of  reinforced   concrete 

for,  6 
Retaining  walls, 

advantages  of  reinforced  concrete 
for,  6,  289 

design  of,  295 

examples  of,  299 

fluid  pressure  on,  290 

proportions  of,  294 

stability  of,  289,  292 

supported  at  top,  301 
Rods, 

spacing  of,  174 

tables  for,  224,  225 

Sand, 

effect  of  size  of  grains,  14 

general  requirements,  9 
Shear  failure  of  beams,  131,  138 
Shear  reinforcement  of  beams,  133 
Shearing  strength  of 

concrete,  17 

T-beams,  144 
Shearing  stress, 

effect  of,  on  tensile  stress,  102 

in  beams,  100,  133,  207 

working  values  of,  173 
Shrinkage  of  concrete,  27 
Shrinkage  stresses,  40 
Spacing  of  rods  or  bars,  174 
Steel, 

coefficient  of  expansion  of,  33 

corrosion  of,  in  concrete,  191 

elongation  of,  32 

for  columns,  189 

general  requirements,  30 

modulus  of  elasticity  of,  32 


Steel, 

quality  of,  32,  172 

tensile  strength  of,  32 

working  stresses  for,  170 
Stirrups,  134 
Stress-strain  curves,  26 

Tables  for 

floor-slabs,  227,  236 

rods,  224,  225 

Tanks,  advantages  of  reinforced  con- 
crete for,  7 
T-beams, 

double  reinforced,  84 

economical  proportions  for,  184 

formulas  for,  79,  201 

notation  for,  79,  201 

shearing  strength  of,  144 

strength  of,  142 

tests  of,  142 
Temperature  stresses,  40,  194 

in  arches,  272 
Tests  of 

adhesion,  34 

beams,  120,  138 

columns,  151,  153,  159 

T-beams,  142 
Trestles, 

analysis  of  stresses  in,  305 

examples  of,  309 

Unit  frames,  257 

Walls,  262 

Web  reinforcement,  133,  173 
Working  stresses,  166,  211 
Working  stresses  for 

beams,  170 

bond, 172 

columns,  185,  221 

concrete,  170 

steel,  170 


^v 

THE  \ 

UNIVERSITY  } 

/ 


AN  INITIAL  FINE  OF  25  CENTS 


*  OO  I 


OP.  CALIFORNIA  UBRARY 


